Calculus: Derivative Rules: Power, Chain, Product, Quotient Worksheet

Question 1

Find d/dx of x^7.

Accepted Answer

The derivative is 7x^6 by the power rule.

Question 2

Let f(x) = 5x^4 - 3x^2 + 7. Find f'(x).

Accepted Answer

The derivative is f'(x) = 20x^3 - 6x. The constant 7 has derivative 0.

Question 3

Find the derivative of y = 4x^(-3) + sqrt(x).

Accepted Answer

The derivative is y' = -12x^(-4) + 1/(2sqrt(x)). This uses the power rule with sqrt(x) written as x^(1/2).

Question 4

Find the derivative of y = (3x^2 - 1)^5.

Accepted Answer

The derivative is y' = 30x(3x^2 - 1)^4. The outside derivative gives 5(3x^2 - 1)^4, and the inside derivative is 6x.

Question 5

Find the derivative of y = sqrt(2x + 5).

Accepted Answer

The derivative is y' = 1/sqrt(2x + 5). This comes from writing the function as (2x + 5)^(1/2) and applying the chain rule.

Question 6

Let y = (x^2 + 1)(x^3 - 4x). Find y' using the product rule.

Accepted Answer

The derivative is y' = 2x(x^3 - 4x) + (x^2 + 1)(3x^2 - 4), which simplifies to y' = 5x^4 - 9x^2 - 4.

Question 7

Find the derivative of y = x^2 sin(x).

Accepted Answer

The derivative is y' = 2x sin(x) + x^2 cos(x). This uses the product rule with u = x^2 and v = sin(x).

Question 8

Let y = (2x + 1)/(x - 3). Find y' using the quotient rule.

Accepted Answer

The derivative is y' = -7/(x - 3)^2. The numerator becomes 2(x - 3) - (2x + 1), which simplifies to -7.

Question 9

Find the derivative of y = (x^2 + 4)/(3x^2 - 1).

Accepted Answer

The derivative is y' = -26x/(3x^2 - 1)^2. This comes from applying the quotient rule and simplifying the numerator.

Question 10

Let y = (x^3 - 2x)^4. Find y' and then evaluate y' at x = 1.

Accepted Answer

The derivative is y' = 4(x^3 - 2x)^3(3x^2 - 2). At x = 1, y' = -4.

Question 11

Let y = (x^2 + 1)(3x - 2)^4. Find y'.

Accepted Answer

The derivative is y' = 2x(3x - 2)^4 + 12(x^2 + 1)(3x - 2)^3. This uses the product rule and the chain rule.

Question 12

Find the derivative of y = (x^2 - 1)/(x^2 + 1).

Accepted Answer

The derivative is y' = 4x/(x^2 + 1)^2. The quotient rule gives a numerator of 2x(x^2 + 1) - 2x(x^2 - 1), which simplifies to 4x.

Question 13

Find the equation of the tangent line to f(x) = x^3 - 4x at x = 2.

Accepted Answer

The derivative is f'(x) = 3x^2 - 4, so the slope at x = 2 is 8. Since f(2) = 0, the tangent line is y = 8x - 16.

Question 14

A student says the derivative of y = (x^2 + 1)^3 is y' = 3(x^2 + 1)^2. Identify the mistake and give the correct derivative.

Accepted Answer

The student forgot to multiply by the derivative of the inside function, x^2 + 1. The correct derivative is y' = 6x(x^2 + 1)^2.

Question 15

The position of an object is s(t) = t^4 - 2t^3 + t. Find the velocity v(t), the acceleration a(t), and their values at t = 2.

Accepted Answer

The velocity is v(t) = 4t^3 - 6t^2 + 1, and the acceleration is a(t) = 12t^2 - 12t. At t = 2, the velocity is 9 and the acceleration is 24.

Calculus: Derivative Rules: Power, Chain, Product, Quotient