Limits and Derivatives Miscellaneous Exercise
Question:
Find derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : (x^2+1)cos x
Answer:
 Use the chain rule to calculate the derivative:
d/dx[(x^2 + 1)cos x] = d/dx[(x^2 + 1)]cos x + (x^2 + 1)d/dx[cos x]
 Calculate the derivative of the first term:
d/dx[(x^2 + 1)] = 2x
 Calculate the derivative of the second term:
d/dx[cos x] = sin x
 Substitute the derivatives into the equation:
d/dx[(x^2 + 1)cos x] = 2xcos x  (x^2 + 1)sin x
Question:
Find the derivative of the following functions from first principle: cos(x−π/8)
Answer:
Step 1: Identify the function, f(x) = cos(x−π/8).
Step 2: Use the definition of the derivative to calculate the derivative of f(x):
df/dx = lim h→0 (cos(xπ/8 + h)  cos(xπ/8))/h
Step 3: Use the sum and difference identities for cosine to simplify the equation:
df/dx = lim h→0 (cos(x+hπ/8)  cos(xπ/8))/h
Step 4: Use the product to sum identity for cosine to further simplify the equation:
df/dx = lim h→0 (cos(xπ/8)cos(h)  cos(xπ/8)cos(h))/h
Step 5: Simplify the equation:
df/dx = lim h→0 (cos(xπ/8)(1  cos(h))/h
Step 6: Use the power reduction identity for cosine to simplify the equation:
df/dx = lim h→0 (cos(xπ/8)(2sin2(h/2))/h
Step 7: Use the double angle identity for sin to simplify the equation:
df/dx = lim h→0 (cos(xπ/8)(2sin(h)cos(h))/h
Step 8: Use the limit definition of the derivative to calculate the derivative of f(x):
df/dx = lim h→0 (2sin(h)cos(xπ/8))/h = 2cos(xπ/8)lim h→0 (sin(h))/h = 2cos(xπ/8)·1 = 2cos(xπ/8)
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers : (x+a)
Answer:
Answer:

First, use the power rule: Derivative of (x+a) = d/dx (x+a) = 1

Since a is a constant, the derivative is equal to 1. Therefore, the derivative of (x+a) = 1
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,r and s are fixed nonzero constants and m and n are integers) : a/x^4−b/x^2+cos x
Answer:
Answer: Derivative of a/x^4 = 4a/x^5 Derivative of b/x^2 = 2b/x^3 Derivative of cos x = sin x
Therefore, the derivative of the given function is 4a/x^5 2b/x^3 sin x.
Question:
Find the derivative of the following functions(it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : cosx/(1+sin x)
Answer:
Step 1: Rewrite the given function as cos x/1+sin x
Step 2: Use the quotient rule to find the derivative of the function
Derivative = (1+sin x)(cos x)  (cos x)(sin x) / (1+sin x)^2
Step 3: Simplify the derivative
Derivative = cos^2 x  sin^2 x / (1+sin x)^2
Step 4: Use the double angle identity to simplify the derivative
Derivative = 1 / (1+sin x)^2
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : cosec x cot x
Answer:
Step 1: Use the product rule to find the derivative of cosec x cot x.
Step 2: Differentiate cosec x with respect to x.
Step 3: Differentiate cot x with respect to x.
Step 4: Multiply the derivatives of cosec x and cot x.
Step 5: Simplify the expression.
Answer: The derivative of cosec x cot x is cosec x cosec x.
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : sin^n x
Answer:

Use the chain rule: d/dx (sin^n x) = n sin^(n1) x cos x

Simplify: n sin^(n1) x cos x
Question:
Find the derivative of the following functions(it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : (ax+b)/(cx+d)
Answer:
Step 1: Rewrite the function in the form of a fraction (ax+b)/(cx+d)
Step 2: Apply the quotient rule
Derivative = [(c(ax+b))  (d(cx+d))]/[(cx+d)^2]
Step 3: Simplify the expression
Derivative = [(acx+bc)  (dcx+bd)]/[(cx+d)^2]
Step 4: Simplify further
Derivative = (adbc)/[(cx+d)^2]
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : (sec x−1)/(sec x+1)
Answer:
Answer:
 First, use the quotient rule to find the derivative of (sec x  1)/(sec x + 1):
d/dx [(sec x  1)/(sec x + 1)] = (sec x + 1)(d/dx sec x)  (sec x  1)(d/dx sec x) / (sec x + 1)^2
 Now, use the chain rule to find the derivative of sec x:
d/dx sec x = sec x tan x (d/dx x)
 Substitute this into the equation from step 1:
d/dx [(sec x  1)/(sec x + 1)] = (sec x + 1)(sec x tan x)(d/dx x)  (sec x  1)(sec x tan x)(d/dx x) / (sec x + 1)^2
 Simplify the equation:
d/dx [(sec x  1)/(sec x + 1)] = (sec x tan x)(d/dx x) / (sec x + 1)^2
 Finally, use the power rule to simplify the equation further:
d/dx [(sec x  1)/(sec x + 1)] = (sec^2 x tan x)(d/dx x) / (sec^2 x + 1)
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : (ax+b)(cx+d)^2
Answer:
Answer:
Step 1: Rewrite the function as: (ax+b)(cx^2 + 2dx + d^2)
Step 2: Apply the product rule: d/dx [(ax+b)(cx^2 + 2dx + d^2)] = (ax+b) * (2cx + 2d) + (cx^2 + 2dx + d^2) * (a)
Step 3: Simplify the expression: d/dx [(ax+b)(cx^2 + 2dx + d^2)] = 2acx^2 + (2ad + 2bc)x + (2bd + ab)
Question:
Find the derivative of the following function from first principle: −x
Answer:
Step 1: Identify the function
Function: f(x) = x
Step 2: Compute the difference quotient
Difference Quotient: (f(x + h)  f(x))/h
Step 3: Simplify the difference quotient
Difference Quotient: ((x + h)  (x))/h
Step 4: Take the limit of the difference quotient as h approaches 0
Limit of Difference Quotient: lim h→0 ((x + h)  (x))/h
Step 5: Simplify the limit
Limit of Difference Quotient: lim h→0 (h)/h
Step 6: Compute the limit
Limit of Difference Quotient: lim h→0 1
Step 7: The derivative of the function is
Derivative of f(x) = x: 1
Question:
Find the derivative of the following functions form first principle: (−x)^(−1)
Answer:
Answer: Step 1: Let f(x) = (−x)^(−1)
Step 2: Find f’(x) using the first principle: f’(x) = lim h>0 [f(x+h)  f(x)]/h
Step 3: Substitute the function f(x) in the equation: f’(x) = lim h>0 [((−x+h)^(−1))  (−x)^(−1)]/h
Step 4: Simplify the equation: f’(x) = lim h>0 [((−1/x+h)^(−1))  (−1/x)^(−1)]/h
Step 5: Simplify further: f’(x) = lim h>0 [1/(−1/x+h)  1/(−1/x)]/h
Step 6: Solve for f’(x): f’(x) = lim h>0 [(x  hx)/(x + hx)(x)]/h
Step 7: Simplify: f’(x) = lim h>0 [(1)/(x + hx)]/h
Step 8: Solve for f’(x): f’(x) = lim h>0 [1/(x + hx)]/h
Step 9: Simplify further: f’(x) = 1/x^2
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : (a+b sin x)/(c+d cos x)
Answer:

Rewrite the function as: (a+b sin x)(cd cos x) / (c+d cos x)(c+d cos x)

Take the derivative of each term in the numerator and denominator: Numerator: (a+b sin x)’ (cd cos x) + (cd cos x)’ (a+b sin x) Denominator: (c+d cos x)’ (c+d cos x)

Simplify the derivatives: Numerator: b cos x (cd cos x)  d sin x (a+b sin x) Denominator: 2d cos x (c+d cos x)

Simplify the fraction: (b cos x (cd cos x)  d sin x (a+b sin x)) / (2d cos x (c+d cos x))
Question:
Find the derivative of the following functions(it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : (px^2+qx+r)/(ax+b)
Answer:
Step 1: Rewrite the function as: (px^2 + qx + r)(ax + b)^1
Step 2: Take the derivative of both sides: 2px(ax + b)^1 + q(ax + b)^1  (px^2 + qx + r)(ax + b)^2(a)
Step 3: Simplify the expression: 2px(ax + b)^1 + q(ax + b)^1 + (px^2 + qx + r)(a)(ax + b)^2
Step 4: Simplify further: 2px(ax + b)^1 + q(ax + b)^1  (pa^2x^2 + qax + ra)(ax + b)^2
Question:
Find the derivative of the following fucctions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : (1+1/x)/(1−1/x)
Answer:
Given, f(x) = (1+1/x)/(1−1/x)
Step 1: Rewrite the function as
f(x) = (1+1/x) (1/1−1/x)
Step 2: Take the derivative of both sides
f’(x) = (1/x) (1/1−1/x) + (1+1/x) (1/x2) (1/1−1/x)
Step 3: Simplify the equation
f’(x) = (1/x2)(1+1/x−1−1/x)
Step 4: Simplify further
f’(x) = (1/x2)(2−1/x)
Step 5: Final answer
f’(x) = (2−1/x)/x2
Question:
Find the derivative of the following functions from first principle: sin(x+1)
Answer:

Rewrite the function as f(x) = sin(x+1)

Take the derivative of f(x) using the definition of the derivative: f’(x) = lim h>0 (sin(x+1+h)  sin(x+1))/h

Simplify the expression: f’(x) = lim h>0 (sin(x+h+1)  sin(x+1))/h

Use the identity sin(a+b) = sin(a)cos(b) + cos(a)sin(b): f’(x) = lim h>0 (sin(x+1)cos(h) + cos(x+1)sin(h)  sin(x+1))/h

Simplify the expression: f’(x) = lim h>0 (cos(x+1)sin(h))/h

Use the identity sin(a) = a when a is small: f’(x) = lim h>0 (cos(x+1)h)/h

Simplify the expression: f’(x) = lim h>0 cos(x+1)

Evaluate the limit: f’(x) = cos(x+1)
Question:
Find the derivative of the following function (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : (ax+b)/(cx+d)
Answer:
Given: f(x) = (ax+b)/(cx+d)
Step 1: Rewrite the function in the form of f(x) = u/v
f(x) = (ax+b)/(cx+d) = (ax+b)/v, where v = (cx+d)
Step 2: Apply the quotient rule to calculate the derivative of the function.
f’(x) = [v(du/dx)  u(dv/dx)]/v^2
Step 3: Substitute the values of u and v in the equation.
f’(x) = [v(a)  (ax+b)(c)]/v^2
Step 4: Simplify the equation.
f’(x) = [cx+d)(a)  (ax+b)(c)]/[(cx+d)^2]
f’(x) = [acx+ad  acx  bc]/[(cx+d)^2]
f’(x) = ad/[(cx+d)^2]
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : (ax^2+sin x)(p+q cos x)
Answer:
Step 1: Use the Product Rule to differentiate the given expression.
Step 2: Differentiate the first factor, (ax^2 + sin x), using the Power Rule.
Step 3: Differentiate the second factor, (p + q cos x), using the Chain Rule.
Step 4: Combine the two derivatives to obtain the final answer.
Answer: (2ax + cos x)(p + q cos x) + (ax^2 + sin x)(q sin x)
Question:
If the derivative of the function 4√x−2 is a/√x. Find the value of a.
Answer:
 Use the definition of a derivative to find a:
a = (4√x  2)’ = 4/2√x
 Substitute x = 1 in the equation to solve for a:
a = 4/2√1
 Simplify to find the value of a:
a = 4
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) :(4x+5 sin x)/(3x+7 cosx)
Answer:
 Rewrite the function in its simplest form:
(4x + 5sin x) / (3x + 7cos x)
 Use the Quotient Rule to find the derivative:
[(3x + 7cos x)(4)  (4x + 5sin x)(7cos x)] / (3x + 7cos x)^2
 Simplify the derivative:
[12x + 28cos x + 28x  35sin x] / (3x + 7cos x)^2
 Simplify further:
[40x + 28cos x  35sin x] / (3x + 7cos x)^2
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) :x/(1+tan x)
Answer:
Step 1: Rewrite the function as x/1+tan x
Step 2: Apply the quotient rule to the function
Step 3: Find the derivative of the numerator
Derivative of numerator = 1
Step 4: Find the derivative of the denominator
Derivative of denominator = sec^2 x
Step 5: Substitute the derivatives of the numerator and denominator into the quotient rule
Derivative of x/(1+tan x) = 1/sec^2 x  x sec^2 x/(1+tan x)^2
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r,s and s are fixed nonzero constants and m and n are integers) : x^4(5 sin x−3 cos x)
Answer:
Answer: Step 1: Take the derivative of x^4 Derivative of x^4 = 4x^3
Step 2: Take the derivative of 5 sin x Derivative of 5 sin x = 5 cos x
Step 3: Take the derivative of 3 cos x Derivative of 3 cos x = 3 (sin x)
Step 4: Combine the derivatives Derivative of x^4 (5 sin x  3 cos x) = 4x^3 (5 cos x  3 (sin x))
Step 5: Simplify Derivative of x^4 (5 sin x  3 cos x) = 20x^3 sin x + 9x^3 cos x
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : sin(x+a)/cosx
Answer:
Step 1: Rewrite the given expression as sin(x+a) * cosx^1.
Step 2: Apply the quotient rule for derivatives.
Step 3: Derivative of sin(x+a) = cos(x+a)
Step 4: Derivative of cosx^1 = cosx^2
Step 5: Multiply the derivatives of the numerator and denominator.
Step 6: Simplify the expression.
Answer: The derivative of sin(x+a)/cosx is cos(x+a) * cosx^2 = sin(x+a)cosx^2.
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : (x+cosx)(x−tanx)
Answer:
 Use the product rule to find the derivative of (x + cosx)(x − tanx):
d/dx [(x + cosx)(x − tanx)] = (x + cosx)d/dx (x − tanx) + (x − tanx)d/dx (x + cosx)
 Find the derivative of the first term:
(x + cosx)d/dx (x − tanx) = (x + cosx)(1 − sec2x)
 Find the derivative of the second term:
(x − tanx)d/dx (x + cosx) = (x − tanx)(1 + cosx)
 Add the two derivatives together:
d/dx [(x + cosx)(x − tanx)] = (x + cosx)(1 − sec2x) + (x − tanx)(1 + cosx)
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constant and m and n are integers) : (ax+b)^n
Answer:
Answer:
Step 1: Use the power rule to find the derivative.
Derivative = n(ax + b)^(n1) * a
Step 2: Simplify the derivative.
Derivative = an(ax + b)^(n1)
Question:
Find the derivative of the following function (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : (x^2cos(π/4))/sin x
Answer:
Given, f(x) = (x^2cos(π/4))/sin x
Step 1: Rewrite the function using the product rule
f(x) = (x^2cos(π/4))/sin x = x^2cos(π/4) . (1/sin x)
Step 2: Apply the product rule
f’(x) = [2xcos(π/4) . (1/sin x)] + [x^2cos(π/4) . (cos x/sin^2 x)]
Step 3: Simplify the expression
f’(x) = [2xcos(π/4) . (1/sin x)] + [x^2cos(π/4) . (cos x/sin^2 x)] = (2xcos(π/4))/sin x  (x^2cos(π/4)cos x)/sin^3 x
Hence, the derivative of the given function is f’(x) = (2xcos(π/4))/sin x  (x^2cos(π/4)cos x)/sin^3 x
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : (px+q)(r/x+s)
Answer:
Step 1: Rewrite the function in terms of multiplication and division:
px + q × (r/x + s)
Step 2: Use the product rule to differentiate the function:
(px + q)’ × (r/x + s) + (px + q) × (r/x + s)'
Step 3: Differentiate each term:
p + q’ × (r/x + s) + (px + q) × (r/x2s)
Step 4: Simplify:
p + q’ × (r/x + s) + (px + q) × (r/x2s)
= p + qr/x2  2qs + pxr/x2  2qsr/x2
Step 5: Simplify further:
p + qr/x2  2qs + pxr/x2  2qsr/x2
= p + qr/x2 + pxr/x2  2qs  2qsr/x2
= p(1 + xr/x2) + q(r/x2  2sr/x2)
= p + qr/x2  2qs/x2
Question:
Find the derivative of the following functions(it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers): 1/(ax^2+bx+c)
Answer:

Rewrite the function in the form of y = (d/dx)(ax^2+bx+c)

Differentiate both sides of the equation with respect to x
y’ = (d/dx)(2ax + b)
 Substitute the values of a, b and c
y’ = (d/dx)(2a*x + b)
 Simplify the expression
y’ = 2a
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : sin(x+a)
Answer:

Use the Chain Rule: d/dx[sin(x+a)] = d/du[sin(u)] * du/dx[x+a]

Differentiate the inside function: d/du[sin(u)] = cos(u)

Differentiate the outside function: du/dx[x+a] = 1

Substitute: d/dx[sin(x+a)] = cos(x+a)
Question:
Find the derivative of the following (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : (sin x+cos x)/(sinx−cosx)
Answer:
Answer: Step 1: Rewrite the expression as: (sin x+cos x)/(sinx−cosx) = (a sin x + b cos x)/(c sin x − d cos x) Step 2: Use the Quotient Rule to find the derivative: d/dx [(a sin x + b cos x)/(c sin x − d cos x)] = (p sin x + q cos x)(c sin x − d cos x)  (a sin x + b cos x)(r sin x + s cos x)/[(c sin x − d cos x)^2] Step 3: Simplify the expression: d/dx [(a sin x + b cos x)/(c sin x − d cos x)] = (p sin x + q cos x)(c sin x − d cos x)  (a sin x + b cos x)(r sin x + s cos x)/[(c sin x − d cos x)^2] = (p c  q d) sin x cos x + (q c  p d) sin^2 x  (a r + b s) sin x cos x  (b r + a s) cos^2 x/[(c sin x − d cos x)^2]
Question:
Find the derivative of the following fuctions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : x/sin^2x
Answer:
Step 1: Rewrite the function as x/sin^2x = x(sin^2x)^1
Step 2: Take the derivative of both sides of the equation using the chain rule: d/dx[x(sin^2x)^1] = (sin^2x)^1d/dx[x] + xd/dx[(sin^2x)^1]
Step 3: Simplify the equation on the right hand side: d/dx[x(sin^2x)^1] = (sin^2x)^11 + x(2sinxcosx)*(sin^2x)^2
Step 4: Simplify the equation further: d/dx[x(sin^2x)^1] = (sin^2x)^1  2xsinxcosx*(sin^2x)^2
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are fixed integers) : (x+secx)(x−tanx)
Answer:
Step 1: Rewrite the function as: (x + secx)*(x  tanx)
Step 2: Use the Product Rule to find the derivative of the function: d/dx [(x + secx)(x  tanx)] = (d/dx [x + secx])(x  tanx) + (x + secx)*(d/dx [x  tanx])
Step 3: Use the Chain Rule to find the derivatives of the individual terms: d/dx [x + secx] = 1 + (d/dx [secx]) d/dx [x  tanx] = 1  (d/dx [tanx])
Step 4: Use the Chain Rule to find the derivatives of secx and tanx: d/dx [secx] = secx*tanx d/dx [tanx] = sec^2x
Step 5: Substitute the derivatives of secx and tanx into the equation: d/dx [(x + secx)(x  tanx)] = (1 + secxtanx)(x  tanx) + (x + secx)(1  sec^2x)
Step 6: Simplify the equation: d/dx [(x + secx)(x  tanx)] = x  tanx + secxtanx + x  secx*sec^2x
Step 7: Simplify the equation further: d/dx [(x + secx)(x  tanx)] = 2x  (tanx + secxsec^2x)
Question:
Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed nonzero constants and m and n are integers) : (ax+b)^n(cx+d)^m
Answer:
Answer:

Use the power rule to expand the function: Derivative of (ax + b)^n = n(ax + b)^(n1) * (a) Derivative of (cx + d)^m = m(cx + d)^(m1) * (c)

Multiply the two derivatives: n(ax + b)^(n1) * (a) * m(cx + d)^(m1) * (c)

Simplify the expression: nm(ax + b)^(n1) * (cx + d)^(m1) * (ac)