# Derivatives of cos, sin, e^x, ln(x)

> AP Calculus AB · Differentiation: Definition and Fundamental Properties
> Source: https://www.owlsprep.com/study/ap-calculus-ab-u2-derivatives-of-cos-sin-e/

This guide covers core derivative rules for four common transcendental functions: sine, cosine, natural exponential $e^x$, and natural logarithm $\ln x$, with worked examples, common pitfalls, and exam tips for AP Calculus AB.

**Prerequisites:** Limit definition of the derivative; Standard trigonometric limit identities; Basic properties of exponential and logarithmic functions; [Unit 2 Derivatives Overview](https://www.owlsprep.com/study/ap-calculus-ab-u2-overview/)

## Learning objectives

- Derive and recall core derivative rules for $\sin x$, $\cos x$, $e^x$, and $\ln x$
- Apply these rules to differentiate combinations of these functions
- Calculate tangent line slopes and equations using these rules
- Identify and avoid common mistakes when applying these rules

## Core Overview

This topic establishes four core derivative rules for the most common non-algebraic (transcendental) functions tested on AP Calculus AB. It makes up part of the 10–12% of the total exam score allocated to Unit 2, and appears in nearly every exam as both standalone multiple choice and a required intermediate step in longer free response questions.

You will use these rules to solve problems asking for derivatives of function combinations, tangent line slopes, and instantaneous rates of change. Unlike the power rule for algebraic functions, these rules are specific to each function and require memorization (rooted in first principles) to apply correctly. Mastery of these rules is non-negotiable for all later differentiation topics.

## Derivatives of Sine and Cosine

**Derivation:** Derive derivative rules for $\sin x$ and $\cos x$ from first principles

*Starting from:* Limit definition of the derivative and trigonometric addition identities

1. Start with the limit definition for $\frac{d}{dx}[\sin x]$:
2. $$\lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h}$$
3. Use the sine addition identity to split the limit:
4. $$\lim_{h \to 0} \left( \sin x \cdot \frac{\cos h - 1}{h} + \cos x \cdot \frac{\sin h}{h} \right)$$
5. Substitute the standard trigonometric limits $\lim_{h \to 0} \frac{\cos h - 1}{h} = 0$ and $\lim_{h \to 0} \frac{\sin h}{h} = 1$, then simplify. Repeating the process for $\cos x$ gives the second rule.

*Conclusion:* The final derivative rules are $\frac{d}{dx}[\sin x] = \cos x$ and $\frac{d}{dx}[\cos x] = -\sin x$ for all real $x$.

**Derivative Rules for Sine and Cosine** — The derivative of $\sin x$ is $\cos x$, and the derivative of $\cos x$ is $-\sin x$. The negative sign for cosine is confirmed by the slope of the cosine graph at any test point.

*Example:* At $x = \frac{\pi}{2}$, the slope of $\cos x$ is $-1$, which matches $-\sin\left(\frac{\pi}{2}\right) = -1$.

**Worked example:** Find the slope of the tangent line to $f(x) = 3\sin x - 2\cos x$ at $x = \pi$.

1. Use the sum rule and constant multiple rule to split differentiation into individual terms.
2. Apply the derivative rules for sine and cosine:

   $$\frac{d}{dx}[3\sin x] = 3\cos x \quad \frac{d}{dx}[-2\cos x] = -2(-\sin x) = 2\sin x$$
3. Combine terms to get the derivative function:

   $$f'(x) = 3\cos x + 2\sin x$$
4. Evaluate the derivative at $x = \pi$:

   $$f'(\pi) = 3\cos(\pi) + 2\sin(\pi) = 3(-1) + 2(0) = -3$$
5. The slope of the tangent line at $x = \pi$ is $-3$.

> **Exam tip:** Always double-check the sign on the derivative of cosine: the negative sign is the most commonly missed detail on trig derivative multiple choice questions. If your answer is off by a negative sign, this is almost always the error.

## Derivative of the Natural Exponential $e^x$

**Derivation:** Derive the derivative of $e^x$ from the limit definition

*Starting from:* Limit definition of derivative and definition of Euler's constant $e$

1. Start with the limit definition:

   $$\frac{d}{dx}[e^x] = \lim_{h \to 0} \frac{e^{x+h} - e^x}{h}$$
2. Factor out $e^x$, which is constant with respect to the limit:

   $$e^x \lim_{h \to 0} \frac{e^h - 1}{h}$$
3. By definition of $e$, the limit $\lim_{h \to 0} \frac{e^h - 1}{h} = 1$.

*Conclusion:* $\frac{d}{dx}[e^x] = e^x$: the natural exponential function is its own derivative.

> **warning**
>
> This rule applies only when the exponent is exactly $x$. You will extend this rule to composite exponential functions when you learn the chain rule, but do NOT apply the power rule to $e^x$, since $e$ is a constant base, not a variable.

**Worked example:** Given $f(x) = 5e^x + \sin x - 4$, find $f'(0)$.

1. Apply the sum/difference rule to differentiate term by term.
2. Differentiate each term using core rules:

   $$\frac{d}{dx}[5e^x] = 5e^x, \quad \frac{d}{dx}[\sin x] = \cos x, \quad \frac{d}{dx}[-4] = 0$$
3. Combine terms to get the derivative function:

   $$f'(x) = 5e^x + \cos x$$
4. Evaluate at $x=0$:

   $$f'(0) = 5e^0 + \cos 0 = 5(1) + 1 = 6$$
5. Final result: $f'(0) = 6$.

> **Exam tip:** Don't confuse the derivative of $e^x$ with the power rule: if you mistakenly apply the power rule to $e^x$, you'll get $xe^{x-1}$, which is incorrect.

## Derivative of the Natural Logarithm $\ln x$

**Derivation:** Derive the derivative of $\ln x$ using implicit differentiation

*Starting from:* Definition of natural logarithm as inverse of $e^x$, derivative of $e^x$

1. Let $y = \ln x$, which by definition gives $e^y = x$, for $x > 0$.
2. Differentiate both sides with respect to $x$:

   $$e^y \frac{dy}{dx} = 1$$
3. Solve for $\frac{dy}{dx}$ and substitute $e^y = x$:

   $$\frac{dy}{dx} = \frac{1}{e^y} = \frac{1}{x}$$

*Conclusion:* The derivative of $\ln x$ is $\frac{1}{x}$, defined only for $x > 0$, matching the domain of the original function.

**Derivative of Natural Logarithm** — For all $x > 0$, $\frac{d}{dx}[\ln x] = \frac{1}{x}$. The derivative is always positive, which matches the fact that $\ln x$ is always increasing on its domain.

**Worked example:** Find the equation of the tangent line to $y = \ln x + 2e^x$ at the point $(1, 2e)$.

1. Differentiate term by term using the rules for $\ln x$ and $e^x$:

   $$\frac{dy}{dx} = \frac{1}{x} + 2e^x$$
2. Calculate the slope at $x=1$ by evaluating the derivative:

   $$m = \frac{1}{1} + 2e^1 = 1 + 2e$$
3. Use point-slope form of a line: $y - y_1 = m(x - x_1)$, with $(x_1, y_1) = (1, 2e)$.
4. Substitute values and simplify to slope-intercept form:

   $$y - 2e = (1 + 2e)(x - 1) \implies y = (1 + 2e)x - 1$$
5. The equation of the tangent line is $y = (1 + 2e)x - 1$.

> **Exam tip:** Always remember the domain restriction for $\ln x$: if a question asks for the derivative of $\ln x$ at a non-positive $x$, the derivative does not exist, because $\ln x$ itself is undefined there. This is a common trick question on multiple choice.

## Concept Check

**Check your understanding**

Test your understanding of the core rules with this AP-style multiple choice question.

1. Which of the following is equal to $f'(x)$ if $f(x) = 2e^x - \ln x + 5\cos x$?

   - A) $2e^x - \frac{1}{x} - 5\sin x$
   - B) $2e^x - \frac{1}{x} + 5\sin x$
   - C) $2e^x + \frac{1}{x} - 5\sin x$
   - D) $2x e^{x-1} - \frac{1}{x} - 5\sin x$

   *Answer:* A) $2e^x - \frac{1}{x} - 5\sin x$

   *Why:* Correct. Differentiating term-by-term gives $\frac{d}{dx}[2e^x] = 2e^x$, $\frac{d}{dx}[-\ln x] = -\frac{1}{x}$, $\frac{d}{dx}[5\cos x] = -5\sin x$, which matches option A. Option D incorrectly uses the power rule for $e^x$.

## Common pitfalls

- **Wrong:** Differentiating $\cos x$ to get $\frac{d}{dx}[\cos x] = \sin x$ (missing the negative sign)
  - Why it fails: Students mix up the order of sine and cosine derivatives, memorizing both as positive.
  - Correct: Always write the derivative of cosine immediately with the negative sign when you start a problem, and check that the sign matches the slope of the cosine graph at a test point like $x = \frac{\pi}{2}$.
- **Wrong:** Applying the power rule to $e^x$ to get $\frac{d}{dx}[e^x] = x e^{x-1}$
  - Why it fails: Students confuse the constant base $e$ with a variable base $x$ in power functions.
  - Correct: When you see $e$ as the base, immediately recall it's an exponential function, so use the rule $\frac{d}{dx}[e^x] = e^x$, not the power rule.
- **Wrong:** Writing the derivative of $\ln x$ as $\frac{d}{dx}[\ln x] = x$ or $\frac{d}{dx}[\ln x] = \ln(1/x)$
  - Why it fails: Students mix up the derivative of $\ln x$ with the derivative of $e^x$ (which is itself), so they incorrectly assume $\ln x$ is also its own derivative.
  - Correct: Always associate the natural log derivative with reciprocal: $\ln x \rightarrow \frac{1}{x}$, not anything else.
- **Wrong:** Attempting to evaluate the derivative of $\ln x$ at $x=0$ or $x=-2$, and getting $\frac{1}{x} = -\frac{1}{2}$ as the slope
  - Why it fails: Students forget the domain of $\ln x$ is only positive $x$, so the function does not exist for non-positive inputs, so its derivative also does not exist there.
  - Correct: Before evaluating the derivative of $\ln x$ at a point, check that the point is in the domain of $\ln x$ (i.e., $x>0$) first.
- **Wrong:** Differentiating $\ln 5$ (a constant) to get $\frac{1}{5}$
  - Why it fails: Students see $\ln$ and automatically apply the derivative rule for $\ln x$, forgetting that $\ln 5$ is a constant number, not a function of $x$.
  - Correct: Always check if the term is a constant before applying any derivative rule; the derivative of any constant is zero.

## Cheatsheet

| Category | Formula | Notes |
| --- | --- | --- |
| Derivative of Sine | $\frac{d}{dx}[\sin x] = \cos x$ | Applies when argument is $x$ |
| Derivative of Cosine | $\frac{d}{dx}[\cos x] = -\sin x$ | Don't forget the negative sign |
| Derivative of $e^x$ | $\frac{d}{dx}[e^x] = e^x$ | Do NOT use power rule; $e$ is constant |
| Derivative of $\ln x$ | $\frac{d}{dx}[\ln x] = \frac{1}{x}$ | Only defined for $x>0$ |
| Constant Multiple Rule | $\frac{d}{dx}[c f(x)] = c f'(x)$ | Constants factor out of derivative |
| Sum/Difference Rule | $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$ | Use for linear combinations of functions |

## What's next

This topic gives you the core derivative rules for the four most common transcendental functions, which you will use in every upcoming differentiation topic in AP Calculus AB. Immediately next, you will learn the product rule and quotient rule for differentiating products and quotients of these basic functions, followed by the chain rule for differentiating composite functions like $e^{2x}$ or $\sin(x^2)$. Without memorizing these four basic derivative rules correctly, you will not be able to correctly apply product, quotient, or chain rule, leading to unnecessary errors on every subsequent differentiation problem. In the bigger picture, these rules are foundational for all later topics including graph analysis, related rates, optimization, and integration.

- [Chain Rule](https://www.owlsprep.com/study/ap-calculus-ab-u3-chain-rule/)
- [Implicit Differentiation](https://www.owlsprep.com/study/ap-calculus-ab-u3-implicit-differentiation/)
- [Product Rule](https://www.owlsprep.com/study/ap-calculus-ab-u2-product-rule/)

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