Limits and Derivatives Exercise 2
Question:
Find the derivative of the following: (5x^3+3x−1)(x−1).
Answer:
(5x^3+3x−1)(x−1)
= (5x^3+3x−1)(1) + (5x^3+3x−1)(1)
= (5x^3+3x−1) + (5x^33x+1)
= 0 + 0
= 0
Derivative of (5x^3+3x−1)(x−1) = 0
Question:
Find the derivative of the following: x^(−4)(3−4x^(−5)).
Answer:
Step 1: Use the power rule to rewrite the expression as: x^(4 + (5))(3  4x^(5))
Step 2: Apply the power rule to the first term: (4 + (5))x^(9)
Step 3: Apply the power rule to the second term: 4(5)x^(6)
Step 4: Combine the results: (9)x^(9)  4(5)x^(6)
Step 5: Simplify: 9x^(9) + 20x^(6)
Question:
Find the derivative of the following functions:2tanx−7secx
Answer:

Use the Chain Rule: d/dx[2tanx  7secx] = (2)(sec^2x)  (7)(secxtanx)

Simplify: (2)(sec^2x)  (7)(secxtanx) = 2sec^2x  7sec^2x = 5sec^2x
Question:
Find the derivative of x at x=1
Answer:
Step 1: Begin with the equation for the derivative of a function f(x): f’(x) = lim (h→0) (f(x+h)  f(x))/h
Step 2: Substitute x=1 into the equation for the derivative of a function f(x): f’(1) = lim (h→0) (f(1+h)  f(1))/h
Step 3: Evaluate the limit as h approaches 0: f’(1) = lim (h→0) ((1+h)  1)/h
Step 4: Simplify the equation: f’(1) = lim (h→0) (h/h)
Step 5: Evaluate the limit as h approaches 0: f’(1) = lim (h→0) 1
Step 6: The answer is: f’(1) = 1
Question:
For some constant a and b, find the derivative of the following functions: (ax^2+b)^2.
Answer:
Answer: Step 1: Use the power rule to find the derivative of (ax^2+b)^2.
Derivative = 2(ax^2+b)^2 * (2ax)
Step 2: Simplify the expression.
Derivative = 4a(ax^2+b)^2x
Question:
Find the derivatives of the following: secx.
Answer:

Use the chain rule: dy/dx = dy/du * du/dx

Identify the function: y = secx

Identify the derivative of the inner function: du/dx = tanx

Substitute into the chain rule: dy/dx = secx * tanx

Simplify: dy/dx = secxtanx
Question:
Find the derivative of the following functions: sin x cos x
Answer:

Use the product rule: d/dx (sin x cos x) = (d/dx sin x) cos x + (d/dx cos x) sin x

Use the chain rule: d/dx (sin x cos x) = (cos x)(d/dx sin x) + (sin x)(d/dx cos x)

Use the derivatives of sin x and cos x: d/dx (sin x cos x) = (cos x)(cos x) + (sin x)(sin x)

Simplify: d/dx (sin x cos x) = cos2x  sin2x
Question:
Find the derivative of x^(−3)(5+3x)
Answer:
Answer: Step 1: Take the natural log of both sides: ln(x^(3)(5+3x))
Step 2: Use the product rule to take the derivative: (3x^(4))(5+3x) + x^(3)(3)
Step 3: Simplify: 3x^(4)(5+3x) + 3x^(3)
Step 4: Simplify further: 15x^(4) + 3x^(3)
Question:
Find the derivative of the following functions: 5secx+4cosx
Answer:
Answer: Step 1: Take the derivative of 5secx Derivative of 5secx = 5secx tanx
Step 2: Take the derivative of 4cosx Derivative of 4cosx = 4sinx
Step 3: Combine the derivatives Derivative of 5secx + 4cosx = 5secx tanx  4sinx
Question:
Find the derivative of the following functions from first principle x^3−27
Answer:
Answer: Step 1: Identify the given function as f(x) = x^3 − 27
Step 2: Take the derivative of the function using the first principle, i.e., using the power rule
d/dx (f(x)) = d/dx (x^3 − 27)
Step 3: Apply the power rule:
d/dx (x^3 − 27) = 3x^2 (derivative of x^3) − 0 (derivative of 27)
Step 4: Simplify the expression:
d/dx (x^3 − 27) = 3x^2
Therefore, the derivative of the given function f(x) = x^3 − 27 is 3x^2.
Question:
For any constant real number a, find the derivative of: x^n+ax^(n−1)+a^2x^(n−2)+…+a^(n−1)x+a^n.
Answer:
Answer: Step 1: Use the power rule of derivatives to find the derivative of each term:
d/dx[x^n] = nx^(n1)
d/dx[ax^(n1)] = (n1)ax^(n2)
d/dx[a^2x^(n2)] = (n2)a^2x^(n3)
d/dx[a^(n1)x] = (n1)a^(n1)
d/dx[a^n] = 0
Step 2: Combine the derivatives to get the derivative of the entire expression:
d/dx[x^n+ax^(n−1)+a^2x^(n−2)+…+a^(n−1)x+a^n] = nx^(n1) + (n1)ax^(n2) + (n2)a^2x^(n3) + … + (n1)a^(n1)
Question:
Find the derivative of the following functions: cosecx
Answer:
Answer: Step 1: Use the chain rule to rewrite the function as: cosecx = 1/sinx
Step 2: Take the derivative of 1/sinx using the quotient rule: d/dx (1/sinx) = cosx/sinx^2
Step 3: Rewrite the expression using the cosecant function: d/dx (cosecx) = cosecx * cotx
Question:
Find the derivative of x^2  2 at x = 10
Answer:

Identify the function: x^2  2

Take the derivative of the function: 2x

Substitute x = 10 into the derivative: 2(10) = 20
Question:
If f(x)=x^100/100+x^99/99+x^98/98 +…+x+1,showthatf’(1)=100 f’(0)$$.
Answer:
f’(x) = 100x^99/100 + 99x^98/99 + 98x^97/98 + … + 1
f’(1) = 100x^99/100 + 99x^98/99 + 98x^97/98 + … + 1
f’(1) = 100
f’(0) = 0x^99/100 + 99x^98/99 + 98x^97/98 + … + 1
f’(0) = 1
Question:
For some constant a and b, find the derivative of the following functions: (x−a)(x−b).
Answer:
Answer: Step 1: Rewrite the function as: (xa)(xb) = x^2  (a+b)x + ab
Step 2: Take the derivative of the function: 2x  (a+b)
Step 3: Simplify the derivative: 2x  (a+b) = 2x  a  b
Question:
Find the derivatives of the following: 5sinx−6cosx+7.
Answer:
Answer:

Use the Chain Rule: d/dx[5sinx−6cosx+7] = d/dx[5sinx]  d/dx[6cosx] + d/dx[7]

Use the derivatives of sine and cosine: d/dx[5sinx] = 5cosx d/dx[6cosx] = 6sinx d/dx[7] = 0

Substitute the derivatives into the original equation: d/dx[5sinx−6cosx+7] = 5cosx  6sinx + 0

Simplify: d/dx[5sinx−6cosx+7] = 5cosx  6sinx
Question:
Find the derivative of 2x−3/4
Answer:
Answer:
 Rewrite the expression in terms of a fraction: (2x  3)/4
 Take the derivative of numerator and denominator separately: Numerator: 2 Denominator: 0
 Combine the two derivatives: 2/0
 Simplify: Undefined
Question:
For some constant a and b, find the derivative of the following functions: (x−a)/(x−b).
Answer:
Answer: Step 1: Rewrite the function as: (x  a)(x  b)^1
Step 2: Use the power rule to take the derivative of the function: 1(x  a)(x  b)^2 + (x  b)^1
Step 3: Simplify the derivative: (b  a)(x  b)^2
Question:
Find the derivative of the following: x^5(3−6x^(−9)).
Answer:
Step 1: Rewrite the expression as x^14  6x^5.
Step 2: Use the power rule to find the derivative.
Step 3: The derivative of x^14 is 14x^13.
Step 4: The derivative of 6x^5 is 30x^4.
Step 5: Combine the two derivatives to get the final answer: 14x^13  30x^4.
Question:
Find the derivative of
Answer:
y = x^2

Rewrite the equation as y’ = (x^2)'

Apply the power rule: (x^2)’ = 2x^(21)

Simplify: (x^2)’ = 2x^1

Rewrite the equation as y’ = 2x
Question:
2/(x+1)−x^2/(3x−1)
Answer:
 Simplify the left side: 2/(x+1)
 Simplify the right side: x^2/(3x1)
 Combine the two sides: 2/(x+1)  (x^2/(3x1))
 Combine the denominators: (3x1)(x+1)
 Multiply both sides by the denominator: (3x1)(x+1) [2/(x+1)  (x^2/(3x1))]
 Simplify the left side: (3x1) + (2x^2)
 Simplify the right side: x^3  2
 Combine the two sides: (3x1) + (2x^2)  (x^3  2)
 Combine like terms: 3x1 + 2x^2 + x^3 + 2
 Simplify: 4x^3 + 3x  1
Question:
Find the derivative of cosx by first principle.
Answer:
Answer: Step 1: Let y = cosx
Step 2: Calculate the derivative of y with respect to x using the first principle:
d/dx(y) = limh→0 (y + h  y) / h
Step 3: Substitute y = cosx into the equation:
d/dx(cosx) = limh→0 (cosx + h  cosx) / h
Step 4: Simplify the equation:
d/dx(cosx) = limh→0 (h) / h
Step 5: The limit of h as it approaches 0 is 0, so the equation becomes:
d/dx(cosx) = 0
Question:
Find the derivative of 99x at x=100
Answer:
Answer: Step 1: Rewrite the equation as y = 99x
Step 2: Take the derivative of y with respect to x: dy/dx = 99
Step 3: Evaluate the derivative at x=100: dy/dx = 99(100) = 9900
Question:
Find the derivatives of the following: 3cotx+5cosecx.
Answer:
Answer:
Step 1: Use the chain rule to find the derivative of 3cotx.
d/dx (3cotx) = 3csc2x
Step 2: Use the chain rule to find the derivative of 5cosecx.
d/dx (5cosecx) = 5cosecxcotx
Therefore, the derivatives of 3cotx+5cosecx is 3csc2x + 5cosecxcotx.
Question:
Find the derivative of (x^n−a^n)/(x−a) for some constant a
Answer:
Answer:

Rewrite the given expression as: (x^n  a^n)/(x  a)

Apply the quotient rule for derivatives: n*(x^(n1))(x  a)  n(a^(n1))/(x  a)^2

Simplify the fraction: n*(x^(n1))  n*(a^(n1))/(x  a)

Factor out the common terms: n*(x^(n1)  a^(n1))/(x  a)