inside another function, e.g., $$f(g(x))$$

main function in a composite, e.g., $$f$$ in $$f(g(x))$$

function inside, e.g., $$g$$ in $$f(g(x))$$

$$y = f(u)$$ and $$u = g(x)$$ , then $$\frac{dy}{dx} = f$$

Example: $$y = (3x+2)^5$$

$$5(3x+2)^4 \cdot 3 = 15(3x+2)^4$$

Example: $$y = \sin(2x)$$

$$\cos(2x) \cdot 2 = 2\cos(2x)$$

Example: $$y = e^{x^2}$$

$$e^{x^2} \cdot 2x = 2x e^{x^2}$$

Example: $$y = \ln(5x+1)$$

$$\frac{1}{5x+1} \cdot 5 = \frac{5}{5x+1}$$

Example: $$y = \sqrt{4x-7}$$

$$\frac{1}{2\sqrt{4x-7}} \cdot 4 = \frac{2}{\sqrt{4x-7}}$$

Practice: $$y = (2x-1)^3$$

$$3(2x-1)^2 \cdot 2 = 6(2x-1)^2$$

Practice: $$y = \cos(5x)$$

$$-\sin(5x) \cdot 5 = -5\sin(5x)$$

Practice: $$y = \tan(x^2)$$

$$\sec^2(x^2) \cdot 2x = 2x\sec^2(x^2)$$

Practice: $$y = \ln(x^3+1)$$

$$\frac{1}{x^3+1} \cdot 3x^2 = \frac{3x^2}{x^3+1}$$

Practice: $$y = \sqrt{3x^2+2}$$

$$\frac{1}{2\sqrt{3x^2+2}} \cdot 6x = \frac{3x}{\sqrt{3x^2+2}}$$

Practice: $$y = \ln(\cos x)$$

$$\frac{1}{\cos x} \cdot (-\sin x) = -\tan x$$

Mistake: Forgetting to multiply by inner derivative

multiply by $$g$$ after differentiating $$f(g(x))$$

Mistake: Treating composite as product

rule is not the product rule; use correct formula

Mistake: Incorrect identification of inner/outer

$$f$$ and $$g$$ in $$f(g(x))$$ before differentiating

Mistake: Dropping parentheses

expressions grouped, e.g., $$(3x+2)^4$$ , not $$3x+2^4$$

Application: Physics (velocity)

rule for related rates, e.g., $$\frac{ds}{dt} = \frac{ds}{dx} \cdot \frac{dx}{dt}$$

Application: Economics (cost functions)

rule for marginal cost when cost depends on demand, which depends on price

Application: Biology (population models)

rule for rates of change in population with respect to time

Visualization: Function composition diagram

latex="x \rightarrow g(x) \rightarrow f(g(x))">

Visualization: Slope transformation

rule as stretching/compressing slopes via inner function

Visualization: Derivative tree

for outer and inner derivatives, multiply at the end

Mastering the Chain Rule in Calculus Flashcards

Student preview

30 questions

Show all answers

1.

FLASHCARD QUESTION

Front

Chain Rule Definition

Back

2.

FLASHCARD QUESTION

Front

Composite Function

Back

3.

FLASHCARD QUESTION

Front

Outer Function

Back

4.

FLASHCARD QUESTION

Front

Inner Function

Back

5.

FLASHCARD QUESTION

Front

Chain Rule Notation

Back

6.

FLASHCARD QUESTION

Front

Back

7.

FLASHCARD QUESTION

Front

Back

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