AP Calculus AB · September 23, 2026

The Chain Rule

A function inside a function needs another derivative factor. Learn to identify the layers, work from the outside inward, and extend the rule to exponential functions.

40-minute edited lesson12 chaptersCaptions + corrected transcript

01 · Watch

Lesson video

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0:0001The Chain Rule 1:4702Inside and Outside Functions 3:5103The Composite Function Rule 4:4304Why the Chain Rule Works 10:4305Sine of a Polynomial 12:0806Leibniz Notation and More Layers 17:2307Exponential and Power Examples 21:0908Combine Product and Chain Rules 25:0709Three Nested Functions 29:4610Exponential Bases Other Than e 36:0811General Exponential Rule 37:1212An Exponential with a Secant Exponent

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02 · Understand

Learning targets

Identify the inside and outside functions in a composite expression.
Apply $\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$ without forgetting the inside derivative.
Use Leibniz notation to keep track of three or more nested functions.
Combine the chain rule with product, quotient, and power rules.
Differentiate exponential functions with a positive base other than $e$ .

03 · Connect

Keep the layers connected

These forms express the same outside-to-inside relationship.

01Composite function: $(f\circ g)(x)=f(g(x))$ The output of the inside function becomes the input of the outside function.
02Chain rule: $\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$ The outside derivative is evaluated at the original inside function.
03Leibniz notation: $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$ When $y=f(u)$ and $u=g(x)$ , the two rates multiply. This is a derivative relationship, not ordinary fraction cancellation.
04More layers: $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dv}\frac{dv}{dx}$ Use one derivative factor for each successive layer.
05General exponential: $\frac{d}{dx}b^x=b^x\ln b\qquad(b>0)$ $\frac{d}{dx}b^{u(x)}=b^{u(x)}\ln b\,u'(x)$ For a composite exponent, include its derivative. The formula also gives zero for the constant base-one function.

04 · Apply

Worked examples

Examples from the board, written with clear typeset mathematics.

Two layers

Sine of a polynomial

y=\sin(3x^3-5x+6)

The outside function is sine; the inside function is the polynomial.

Differentiate sine while keeping its input unchanged, then multiply by the derivative of the polynomial.

y'=\cos(3x^3-5x+6)(9x^2-5)

Exponential

An exponential with a cosine exponent

y=e^{\cos x}

The derivative of the outside exponential has the same exponential expression.

Multiply by the derivative of cosine, which is negative sine.

y'=-\sin x\,e^{\cos x}

Power

A whole expression raised to the fifth power

y=(x^4+\sin x)^5

Bring down the 5 and reduce the outside power to 4.

Keep the entire inside expression, then multiply by its derivative.

y'=5(x^4+\sin x)^4(4x^3+\cos x)

Combine rules

Recover a quotient derivative

y=\frac{x^4}{\sin x}=x^4(\sin x)^{-1}

Rewrite the quotient as a product with a negative power.

Use the product rule, and use the chain rule when differentiating the reciprocal of sine. Combine over a common denominator.

y'=\frac{4x^3\sin x-x^4\cos x}{\sin^2x}

Three layers

Sine of an exponential of a polynomial

y=\sin\!\left(e^{x^2-5x}\right)

Work from the outside inward: sine, then the exponential, then the polynomial.

Multiply the three derivative factors; do not skip the innermost derivative.

y'=\cos\!\left(e^{x^2-5x}\right)e^{x^2-5x}(2x-5)

Change the base

Differentiate seven to the x

7^x=e^{x\ln7}

Rewrite the function as an exponential with base e, using the inverse relationship between the exponential and natural logarithm.

The derivative of x times ln 7 is the constant ln 7.

\frac{d}{dx}7^x=7^x\ln7

One more layer

A secant function in the exponent

y=8^{\sec x}

Keep the exponential and multiply by the natural logarithm of its base.

Multiply once more by the derivative of the exponent: secant times tangent.

y'=8^{\sec x}\ln8\,\sec x\tan x

Constant multiple

Keep the coefficient

y=5\cdot10^x\quad\Longrightarrow\quad y'=5\cdot10^x\ln10

The 5 is a constant multiplier; the logarithm factor comes from the exponential base, 10.

05 · Avoid

Common mistakes

Forgetting the inside derivative.

$\frac{d}{dx}\sin(x^2)=2x\cos(x^2)$ . The extra factor is essential.

Changing the inside too early.

Keep the original input inside the outside derivative; multiply by its derivative separately.

Treating composition as multiplication.

$f(g(x))$ and $f(x)g(x)$ have different structures and use different rules.

Skipping a layer.

A nested sine, exponential, and polynomial requires three connected derivative factors.

Using the power rule on an exponential.

$x^7$ has a variable base; $7^x$ has a variable exponent.

Dropping the logarithm factor.

$\frac{d}{dx}7^x=7^x\ln7$ . Only the base $e$ has logarithm factor 1.

06 · Check yourself

Try it first

Reveal each answer after identifying the inside and outside functions.

1Differentiate

\sin(3x^3-5x+6)

$y'=\cos(3x^3-5x+6)(9x^2-5)$

2Differentiate

e^{\cos x}

$y'=-\sin x\,e^{\cos x}$

3Which three layers appear in

y=\sin\!\left(e^{x^2-5x}\right)

From the outside inward: sine, an exponential with base $e$ , and the polynomial $x^2-5x$ .

4Differentiate

7^x

$7^x\ln7$

5Differentiate

8^{\sec x}

$y'=8^{\sec x}\ln8\,\sec x\tan x$

07 · Revisit

Transcript and practice

Search the corrected transcript for a rule or worked example. Captions follow the edited video timeline.

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Return to Wednesday 9/23 in eKadence for the assigned Blue Book Section 2.4 practice.