Derivatives and Differentiation Rules Worksheet

Question 1

Find the derivative of f(x) = 7.

Accepted Answer

The derivative is f'(x) = 0 because the derivative of a constant is always 0.

Question 2

Find the derivative of f(x) = x^5.

Accepted Answer

The derivative is f'(x) = 5x^4 by the power rule.

Question 3

Find the derivative of f(x) = 4x^3 - 2x + 9.

Accepted Answer

The derivative is f'(x) = 12x^2 - 2 because the derivative of 4x^3 is 12x^2, the derivative of -2x is -2, and the derivative of 9 is 0.

Question 4

Find the derivative of f(x) = 6x^4 + x^2 - 8x + 1.

Accepted Answer

The derivative is f'(x) = 24x^3 + 2x - 8.

Question 5

Find the derivative of f(x) = 3x^2(x + 5).

Accepted Answer

The derivative is f'(x) = 6x(x + 5) + 3x^2(1), which simplifies to 9x^2 + 30x. This uses the product rule.

Question 6

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

Accepted Answer

The derivative is f'(x) = 2x(x^3 - 4) + (x^2 + 1)(3x^2). This simplifies to 5x^4 + 3x^2 - 8x.

Question 7

Find the derivative of f(x) = (2x + 1)/(x - 3).

Accepted Answer

The derivative is f'(x) = ((2)(x - 3) - (2x + 1)(1)) / (x - 3)^2. This simplifies to -7/(x - 3)^2.

Question 8

Find the derivative of f(x) = (x^2 + 4)/x.

Accepted Answer

The derivative is f'(x) = (2x(x) - (x^2 + 4)(1)) / x^2, which simplifies to (x^2 - 4)/x^2. An equivalent form is 1 - 4/x^2.

Question 9

Find the derivative of f(x) = (3x - 2)^4.

Accepted Answer

The derivative is f'(x) = 4(3x - 2)^3(3), which simplifies to 12(3x - 2)^3. This uses the chain rule.

Question 10

Find the derivative of f(x) = (x^2 + 5x)^3.

Accepted Answer

The derivative is f'(x) = 3(x^2 + 5x)^2(2x + 5).

Question 11

Find the derivative of f(x) = sqrt(x), written as x^(1/2).

Accepted Answer

The derivative is f'(x) = (1/2)x^(-1/2), which can also be written as 1/(2sqrt(x)).

Question 12

Find the derivative of f(x) = 1/x^3, written as x^(-3).

Accepted Answer

The derivative is f'(x) = -3x^(-4), which can also be written as -3/x^4.

Question 13

A particle moves so that its position is s(t) = 2t^3 - 9t^2 + 12t. Find the velocity function v(t).

Accepted Answer

The velocity function is v(t) = s'(t) = 6t^2 - 18t + 12 because velocity is the derivative of position.

Question 14

The cost function is C(x) = 5x^2 + 10x + 200. Find C'(x) and explain what it represents.

Accepted Answer

The derivative is C'(x) = 10x + 10. It represents the rate at which cost changes with respect to the number of items produced, also called the marginal cost.

Question 15

Find the derivative of f(x) = x^3 - 4x^2 + 7x - 6, then evaluate f'(2).

Accepted Answer

First, f'(x) = 3x^2 - 8x + 7. Then f'(2) = 3(4) - 8(2) + 7 = 12 - 16 + 7 = 3.

Derivatives and Differentiation Rules

Derivatives and Differentiation Rules

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