Calculus · Step-by-step guide

Derivative Rules

Learn the constant, power, sum, product, quotient, and chain rules with examples.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

Differentiate term by term for sums. Use the power rule ddx(xn)=nxn−1\displaystyle \frac{d}{dx}(x^n)=nx^{n-1}, but switch to product, quotient, or chain rules when functions are multiplied, divided, or composed.

ddx(xn)=nxn−1\displaystyle \frac{d}{dx}(x^n)=nx^{n-1}
01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Classify the outer operation

A sum uses termwise rules; a product, quotient, or composition needs its dedicated rule.

2

Constants behave differently

The derivative of a constant is 0, while a constant multiple stays in front.

3

Simplify strategically

Algebraic rewriting can turn a quotient or radical into powers and make differentiation cleaner.

02

See the structure

A picture makes the relationships easier to remember.

tangent line
The derivative at the marked point is the slope of the highlighted tangent line.
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Power and sum

Differentiate 3x4−5x+7\displaystyle 3x^4-5x+7.

  1. Apply the power rule to 3x⁴.
  2. The derivative of −5x is −5.
  3. The constant 7 differentiates to 0.
Answer12x3−5\displaystyle 12x^3-5
Example 2Rewrite first

Differentiate x+2x\displaystyle \sqrt{x}+\frac{2}{x}.

  1. Rewrite as x1/2+2x−1\displaystyle x^{1/2}+2x^{-1}.
  2. Apply the power rule to each term.
  3. Simplify.
Answer12x−2x2\displaystyle \frac{1}{2\sqrt{x}}-\frac{2}{x^2}
Example 3Trig and exponential

Differentiate ex+sin⁡x\displaystyle e^x+\sin x.

  1. The derivative of eˣ is eˣ.
  2. The derivative of sine is cosine.
  3. Add the results.
Answerex+cos⁡x\displaystyle e^x+\cos x
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Lowering an exponent without multiplying by the old exponent.

  • Using the power rule directly on a composite base.

  • Applying a nonexistent quotient-of-derivatives rule.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Derivative Rules

Build fluency across the core derivative formulas.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

Differentiate f(x)=5x4−3x2+7\displaystyle f(x)=5x^4-3x^2+7.

Show hint

Apply the power rule term by term.

Show worked solution
  1. ddx(5x4)=20x3\displaystyle \frac{d}{dx}(5x^4)=20x^3.
  2. ddx(−3x2)=−6x\displaystyle \frac{d}{dx}(-3x^2)=-6x and the constant derivative is 0.

Answer: f′(x)=20x3−6x\displaystyle f'(x)=20x^3-6x

2

Differentiate g(x)=4x3\displaystyle g(x)=\frac{4}{x^3}.

Show hint

Rewrite with a negative exponent.

Show worked solution
  1. g(x)=4x−3\displaystyle g(x)=4x^{-3}.
  2. Use the power rule: 4(−3)x−4\displaystyle 4(-3)x^{-4}.

Answer: g′(x)=−12x4\displaystyle g'(x)=-\frac{12}{x^4}

3

Differentiate h(x)=x+2x\displaystyle h(x)=\sqrt{x}+2x.

Show hint

Rewrite the root as x1/2\displaystyle x^{1/2}.

Show worked solution
  1. ddx(x1/2)=12x−1/2\displaystyle \frac{d}{dx}(x^{1/2})=\frac{1}{2}x^{-1/2}.
  2. ddx(2x)=2\displaystyle \frac{d}{dx}(2x)=2.

Answer: h′(x)=12x+2\displaystyle h'(x)=\frac{1}{2\sqrt{x}}+2

4

Differentiate p(x)=(x2+1)(x3−2)\displaystyle p(x)=(x^2+1)(x^3-2).

Show hint

Use the product rule.

Show worked solution
  1. u′=2x\displaystyle u'=2x and v′=3x2\displaystyle v'=3x^2.
  2. p′=2x(x3−2)+(x2+1)3x2\displaystyle p'=2x(x^3-2)+(x^2+1)3x^2.

Answer: 5x4+3x2−4x\displaystyle 5x^4+3x^2-4x

5

Differentiate r(x)=sin⁡x+ex\displaystyle r(x)=\sin x+e^x.

Show hint

Use standard derivatives.

Show worked solution
  1. ddx(sin⁡x)=cos⁡x\displaystyle \frac{d}{dx}(\sin x)=\cos x.
  2. ddx(ex)=ex\displaystyle \frac{d}{dx}(e^x)=e^x.

Answer: r′(x)=cos⁡x+ex\displaystyle r'(x)=\cos x+e^x

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

Differentiate term by term for sums. Use the power rule ddx(xn)=nxn−1\displaystyle \frac{d}{dx}(x^n)=nx^{n-1}, but switch to product, quotient, or chain rules when functions are multiplied, divided, or composed.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

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