Continuity and Differentiability Exercise 02
Question:
Differentiate the function with respect to xcosx^3.sin^2(x^5)
Answer:
Answer:
Step 1: Differentiate the function with respect to x:
f’(x) = cosx^3.sin^2(x^5)
Step 2: Differentiate the function with respect to cosx^3:
f’(cosx^3) = sin^2(x^5)
Step 3: Differentiate the function with respect to sin^2(x^5):
f’(sin^2(x^5)) = 2sin(x^5)cos(x^5)
Question:
Differentiate the function with respect to x 2√cot(x^2)
Answer:
Step 1: Take the derivative of the inside function, cot(x^2), with respect to x.
Step 2: Apply the chain rule to the outside function, 2√, with respect to x.
Step 3: Multiply the derivatives of the inside and outside functions together.
Answer: 4x√cot(x^2) csc^2(x^2)
Question:
Differentiate the function with respect to x sec(tan(√x))
Answer:
Step 1: Rewrite the function as sec(arctan(√x))
Step 2: Use the Chain Rule and differentiate with respect to x:
d/dx sec(arctan(√x)) = sec(arctan(√x))sec2(arctan(√x))(1/2)*(1/x^(1/2))
Step 3: Simplify the expression:
d/dx sec(arctan(√x)) = (1/2)*sec(arctan(√x))*sec2(arctan(√x))*x^(1/2)
Question:
Differentiate the function with respect to x sin(ax+b)
Answer:

Differentiate the function: d/dx[sin(ax+b)]

Use the chain rule: d/dx[sin(ax+b)] = d/du[sin(u)] * d/dx[ax+b]

d/du[sin(u)] = cos(u)

d/dx[ax+b] = a

Substitute: d/dx[sin(ax+b)] = a*cos(ax+b)
Question:
Differentiate the function with respect to x cos(√x)
Answer:
Given, f(x) = cos(√x)
Step 1: Take the derivative of f(x) with respect to x.
f’(x) = sin(√x) × (1/2√x)
Step 2: Substitute the value of f’(x) in the given equation.
Differentiate the function with respect to x cos(√x) = sin(√x) × (1/2√x)
Question:
Differentiate the function with respect to x sin(ax+b)/cos(cx+d)
Answer:

(a*cos(ax+b)*cos(cx+d)sin(ax+b)csin(cx+d))/(cos(cx+d))^2

(a*cos(ax+b)*cos(cx+d)sin(ax+b)csin(cx+d))/(cos(cx+d))^2 * (dcos(cx+d)csin(cx+d))

(a*cos(ax+b)*dcos(cx+d)sin(ax+b)ccos(cx+d)asin(ax+b)csin(cx+d)+csin(ax+b)*sin(cx+d))/(cos(cx+d))^3
Question:
Differentiate the function with respect to x sin(x^2+5)
Answer:

Differentiate the inside of the function with respect to x: (2x)

Multiply the result by the outside of the function: (2x)sin(x^2+5)

Simplify: 2xcos(x^2+5)
Question:
Differentiate the function with respect to x cos(sinx)
Answer:
Step 1: Take the derivative of cos(sinx) with respect to x.
Step 2: Use the Chain Rule:
d/dx[cos(sinx)] = sin(sinx) * d/dx[sinx]
Step 3: Take the derivative of sinx with respect to x.
d/dx[sinx] = cosx