AP Calculus AB · Lesson review · September 11, 2026

Derivatives of Sine, Cosine, and e^x

Build the rules from the derivative definition, then combine them in a mixed example.

38-minute edited lesson9 chaptersCaptions + corrected transcript

01 · Watch

Lesson video

0:0001Derivatives of Sine, Cosine, and e^x 0:0502Warm-up: A Tangent Line from the Power Rule 3:5003Derive the Sine Rule 13:1204Derive the Cosine Rule 17:5505Why Tangent Must Wait 21:1506Where the Number e Comes From 25:2607Derive the Exponential Rule 33:3408Combine the New Rules 36:1009Product and Quotient Rule Preview

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02 · Understand

Learning targets

Use the derivative definition and angle-addition identities to derive the sine and cosine rules.
Differentiate sine, cosine, and natural exponential expressions.
Explain why tangent needs the quotient rule, which comes next.
Explain why the special base $e$ makes an exponential function equal to its own derivative.
Combine the new rules with power, constant-multiple, and sum rules.

03 · Rules

The new derivative rules

The two special trigonometric limits make the definition-based proofs work.

01Sine: $\frac{d}{dx}(\sin x)=\cos x$ Differentiate sine to cosine.
02Cosine: $\frac{d}{dx}(\cos x)=-\sin x$ Differentiate cosine to negative sine.
03Natural exponential: $\frac{d}{dx}(e^x)=e^x$ The instantaneous growth rate of $e^x$ is the function value itself.
04Special limits: $\lim_{h\to0}\frac{\sin h}{h}=1,\qquad \lim_{h\to0}\frac{\cos h-1}{h}=0$ These limits supply the two numerical factors in the sine and cosine proofs.
05Tangent preview: $\tan x=\frac{\sin x}{\cos x}\quad\Longrightarrow\quad\text{quotient rule needed}$ The lesson correctly stops here: differentiating this quotient needs a rule that has not yet been established.

04 · Derive and apply

Why the rules work

Start from the definition, use an identity, and separate the limit.

Warm-up

Find a tangent line

f(x)=5x^3+4x^2+7x-2

Differentiate term by term, evaluate the slope at zero, and pair it with the point on the curve:

f'(0)=7,\quad f(0)=-2,\quad y=7x-2

Definition proof

Derivative of sine

\frac{d}{dx}\sin x=\lim_{h\to0}\frac{\sin(x+h)-\sin x}{h}

Use $\sin(x+h)=\sin x\cos h+\cos x\sin h$ .
Group the expression into a cosine-minus-one limit and a sine-over-angle limit.
Apply the two special limits.

\sin x\!\left(0\right)+\cos x\!\left(1\right)=\cos x

Definition proof

Derivative of cosine

\frac{d}{dx}\cos x=\lim_{h\to0}\frac{\cos(x+h)-\cos x}{h}

Use the cosine addition identity, separate the two limit pieces, and apply the same special limits.

\cos x\!\left(0\right)-\sin x\!\left(1\right)=-\sin x

A special base

Why $e^x$ returns itself

e=\lim_{n\to\infty}\left(1+\frac1n\right)^n=\lim_{h\to0}(1+h)^{1/h}

The derivative definition factors out $e^x$ . The remaining limit equals one because of the defining behavior of $e$ .

\frac{d}{dx}e^x=e^x\lim_{h\to0}\frac{e^h-1}{h}=e^x

Mixed practice

Use several rules at once

f(x)=5x^4+2\sin x-4\cos x+8e^x

Apply the power rule, sine rule, cosine rule, and exponential rule term by term.

f'(x)=20x^3+2\cos x+4\sin x+8e^x

05 · Avoid

Common mistakes

Losing the cosine sign.

The derivative of cosine is $-\sin x$ , not $\sin x$ .

Using degrees.

The special trigonometric limits and these derivative formulas require radian measure.

Skipping the angle-addition identity.

$\sin(x+h)$ and $\cos(x+h)$ must be expanded before the limit separates.

Treating every exponential base like $e$ .

The function $e^x$ is special because its derivative factor is exactly one.

Dropping constant coefficients.

Keep each coefficient while differentiating its function.

Differentiating a product term by term.

A product such as $e^x\sin x$ needs a separate product rule.

Dividing derivatives.

The derivative of a quotient is not the quotient of the derivatives; tangent must wait for the quotient rule.

06 · Check yourself

Quick check

Try each before revealing the answer.

1Differentiate

\sin x

$\cos x$ .

2Differentiate

\cos x

$-\sin x$ . Keep the negative sign.

3Differentiate

7e^x

$7e^x$ .

4Differentiate

3x^4-5\cos x+2e^x

$12x^3+5\sin x+2e^x$ .

5Which special limit supplies the factor one in the sine proof?

$\lim_{h\to0}\frac{\sin h}{h}=1$ .

07 · Revisit

Transcript and practice

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

Download corrected transcript

Class practice: complete the assigned Bluebook Section 2.2 problems and use the review sheets in the Friday 9/11 eKadence activity.