Calculus Practice

Derivatives Practice

Practice derivative notation, the power rule, product rule, quotient rule, chain rule, trig derivatives, tangent lines, and differentiability.

Start Practicing Review Derivatives

Try each problem first, then click Show solution to check your work and see the complete step-by-step solution.

Keep These Rules Nearby

Derivatives Essentials

Power Rule

\[ \frac{d}{dx} \left(x^n\right) = nx^{n-1} \]

Bring down the exponent and subtract one.

Product Rule

\[ (fg)' = f'g+fg' \]

Differentiate one factor at a time.

Chain Rule

\[ \frac{d}{dx} f(g(x)) = f'(g(x))g'(x) \]

Differentiate the outside, then multiply by the derivative of the inside.

Download Practice PDF Review Derivatives

Level 1

Derivative Meaning and Basic Rules

Start with derivative notation, constants, and simple power-rule derivatives.

Problem 1

Derivative Meaning

If \[ f'(3)=7, \] what does the number \(7\) represent?

The value \(f(3)\)
The slope of the tangent line at \(x=3\)
The \(x\)-intercept

Step-by-Step Solution

A derivative evaluated at a point gives the instantaneous rate of change there.

Geometrically, it is the slope of the tangent line.

\[ f'(3)=7 \]
Answer: The tangent-line slope at \(x=3\) is \(7\).

Problem 2

Constant Rule

Differentiate: \[ f(x)=12 \]

Step-by-Step Solution

A constant does not change as \(x\) changes.

\[ \frac{d}{dx}(12)=0 \]
Answer: \[ \boxed{f'(x)=0} \]

Problem 3

Power Rule

Differentiate: \[ f(x)=x^6 \]

Step-by-Step Solution

Apply the power rule:

\[ \frac{d}{dx}(x^n) = nx^{n-1} \] \[ \frac{d}{dx}(x^6) = 6x^5 \]
Answer: \[ \boxed{f'(x)=6x^5} \]

Level 2

Polynomials and Powers

Differentiate term by term and work with negative and fractional exponents.

Problem 4

Polynomial

Differentiate: \[ f(x) = 4x^5-3x^3+7x-9 \]

Step-by-Step Solution

Differentiate each term.

\[ \frac{d}{dx}(4x^5) = 20x^4 \] \[ \frac{d}{dx}(-3x^3) = -9x^2 \] \[ \frac{d}{dx}(7x)=7 \] \[ \frac{d}{dx}(-9)=0 \]
Answer: \[ \boxed{ f'(x) = 20x^4-9x^2+7 } \]

Problem 5

Negative Exponent

Differentiate: \[ f(x)=x^{-3} \]

Step-by-Step Solution

Apply the power rule.

\[ f'(x) = -3x^{-4} \]

With positive exponents,

\[ -3x^{-4} = -\frac{3}{x^4}. \]
Answer: \[ \boxed{ f'(x) = -\frac{3}{x^4} } \]

Problem 6

Radical

Differentiate: \[ f(x)=\sqrt{x} \]

Step-by-Step Solution

Rewrite the square root using an exponent.

\[ \sqrt{x} = x^{1/2} \] \[ f'(x) = \frac12 x^{-1/2} \]

Rewrite using a radical:

\[ \frac12x^{-1/2} = \frac{1}{2\sqrt{x}}. \]
Answer: \[ \boxed{ f'(x) = \frac{1}{2\sqrt{x}} } \]

Level 3

Product and Quotient Rules

Differentiate products and quotients of functions without treating the operations term by term.

Problem 7

Product Rule

Differentiate: \[ y = x^2(x^3+1) \]

Step-by-Step Solution

Use the product rule:

\[ (fg)'=f'g+fg'. \]

Let

\[ f=x^2, \qquad g=x^3+1. \] \[ f'=2x, \qquad g'=3x^2. \]
\[ y' = (2x)(x^3+1) + x^2(3x^2) \] \[ = 2x^4+2x+3x^4 \] \[ = 5x^4+2x \]
Answer: \[ \boxed{ y'=5x^4+2x } \]

Problem 8

Product Rule

Differentiate: \[ y=x^3\sin x \]

Step-by-Step Solution

Let

\[ f=x^3, \qquad g=\sin x. \] \[ f'=3x^2, \qquad g'=\cos x. \]
\[ y' = (3x^2)(\sin x) + x^3(\cos x) \]
Answer: \[ \boxed{ y' = 3x^2\sin x + x^3\cos x } \]

Problem 9

Quotient Rule

Differentiate: \[ y = \frac{x^2+1}{x} \]

Step-by-Step Solution

Use the quotient rule:

\[ \left( \frac{f}{g} \right)' = \frac{ f'g-fg' }{ g^2 }. \]

Here,

\[ f=x^2+1, \qquad g=x. \] \[ f'=2x, \qquad g'=1. \]
\[ y' = \frac{ (2x)(x) - (x^2+1)(1) }{ x^2 } \] \[ = \frac{ 2x^2-x^2-1 }{ x^2 } \] \[ = \frac{x^2-1}{x^2} \]
Answer: \[ \boxed{ y' = \frac{x^2-1}{x^2} } \]

Level 4

Chain Rule and Trig Derivatives

Differentiate composite functions and combine the chain rule with trigonometric derivatives.

Problem 10

Chain Rule

Differentiate: \[ y = (3x^2+1)^5 \]

Step-by-Step Solution

Differentiate the outer power first.

\[ 5(3x^2+1)^4 \]

Then multiply by the derivative of the inside:

\[ \frac{d}{dx}(3x^2+1) = 6x. \]
\[ y' = 5(3x^2+1)^4(6x) \] \[ = 30x(3x^2+1)^4 \]
Answer: \[ \boxed{ y' = 30x(3x^2+1)^4 } \]

Problem 11

Trig Derivative

Differentiate: \[ y = 4\sin x-3\cos x \]

Step-by-Step Solution

Recall:

\[ (\sin x)'=\cos x \] \[ (\cos x)'=-\sin x. \]
\[ y' = 4\cos x - 3(-\sin x) \] \[ = 4\cos x + 3\sin x \]
Answer: \[ \boxed{ y' = 4\cos x+3\sin x } \]

Problem 12

Trig + Chain Rule

Differentiate: \[ y = \sin(4x^2) \]

Step-by-Step Solution

Differentiate the outer sine function.

\[ \sin(u) \longrightarrow \cos(u) \]

Keep the inside unchanged:

\[ \cos(4x^2). \]

Now multiply by the derivative of \(4x^2\).

\[ \frac{d}{dx}(4x^2) = 8x. \]
\[ y' = 8x\cos(4x^2) \]
Answer: \[ \boxed{ y' = 8x\cos(4x^2) } \]

Problem 13

Chain Rule

Differentiate: \[ y = \cos(5x) \]

Step-by-Step Solution

Differentiate cosine:

\[ \cos(u) \longrightarrow -\sin(u). \]

Then multiply by the derivative of \(5x\):

\[ \frac{d}{dx}(5x)=5. \]
\[ y' = -5\sin(5x) \]
Answer: \[ \boxed{ y' = -5\sin(5x) } \]

Level 5

Tangent Lines and Mixed Review

Use derivatives geometrically and combine multiple derivative concepts.

Problem 14

Tangent Slope

For \[ f(x)=x^2, \] find the slope of the tangent line at \(x=3\).

Step-by-Step Solution

Differentiate:

\[ f'(x)=2x. \]

Evaluate at \(x=3\):

\[ f'(3) = 2(3) = 6 \]
Answer: \[ \boxed{m=6} \]

Problem 15

Tangent Line

Find the equation of the tangent line to \[ f(x)=x^2 \] at \(x=2\).

Step-by-Step Solution

First find the derivative.

\[ f'(x)=2x. \]

Find the slope at \(x=2\):

\[ f'(2)=4. \]

Find the point:

\[ f(2)=4. \]

Use point-slope form:

\[ y-4 = 4(x-2) \] \[ y = 4x-4 \]
Answer: \[ \boxed{ y=4x-4 } \]

Problem 16

Differentiability

Consider \[ f(x)=|x|. \] Does \(f'(0)\) exist?

Yes, \(f'(0)=0\)
Yes, \(f'(0)=1\)
No, \(f'(0)\) does not exist

Step-by-Step Solution

The graph of \(y=|x|\) has a sharp corner at \(x=0\).

From the left, the slope is

\[ -1. \]

From the right, the slope is

\[ 1. \]

Since the one-sided slopes do not agree, there is no single tangent slope.

Answer: No. \(f'(0)\) does not exist.

Before You Finish

Derivatives Checklist

  1. Interpret a derivative as an instantaneous rate of change and tangent-line slope.
  2. Know that the derivative of a constant is zero.
  3. Apply the power rule correctly.
  4. Rewrite radicals and denominator powers when useful.
  5. Differentiate polynomials term by term.
  6. Use the product rule for products of functions.
  7. Use the quotient rule for ratios of functions.
  8. Recognize nested functions and use the chain rule.
  9. Know the derivatives of sine, cosine, and tangent.
  10. Use \(f'(a)\) to find a tangent-line slope.
  11. Use the derivative and point-slope form to find a tangent-line equation.
  12. Recognize corners and other points where a derivative may not exist.

Continue Learning

Related Calculus Resources

Review derivative rules, revisit limits, or continue into applications of derivatives.

Derivatives Guide

Review derivative meaning, notation, rules, tangent lines, and differentiability.

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Review the limit concepts that lead to the derivative definition.

Applications of Derivatives

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