Derivatives of cos, sin, e^x, ln(x) — AP Calculus BC Study Guide

For: AP Calculus BC candidates sitting AP Calculus BC.

Covers: This chapter covers the derivative definitions and standard formulas for $\sin x$, $\cos x$, the natural exponential function $e^x$, and the natural logarithm $\ln (x)$, including derivations from first principles, routine differentiation, and application to slope and rate problems.

You should already know: Limit evaluation for standard trigonometric limits, definition of the derivative as a limit, algebraic properties of exponential and logarithmic functions.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the AP Calculus BC style for educational use. They are not reproductions of past College Board / Cambridge / IB papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official mark schemes for grading conventions.

1. What Is Derivatives of cos, sin, e^x, ln(x)?

This topic is a core building block of all differentiation, falling in Unit 2: Differentiation: Definition and Fundamental Properties of the AP Calculus CED, which accounts for 10-12% of the total exam weight. It introduces the four most widely used non-polynomial derivatives that appear in every subsequent unit of AP Calculus, from the product rule through related rates to integration by parts. These derivatives are almost always tested in combination with other differentiation rules, but they can also appear as standalone multiple-choice questions or as early, low-difficulty parts of free-response questions that ask for tangent slopes or instantaneous rates of change. Unlike the power rule for polynomials, these derivatives rely on unique limit results that cannot be derived from polynomial differentiation rules, so correct memorization is non-negotiable. This topic appears in both MCQ and FRQ sections: it typically acts as a quick recall check in MCQ, and as a foundational first step for longer multi-part FRQ involving graph analysis or real-world rates.

2. Derivatives of Sine and Cosine

The derivatives of $\sin x$ and $\cos x$ are derived directly from the limit definition of the derivative, using the two key trigonometric limits you already learned: ${\lim }_{h\to 0}\frac{\sin h}{h}=1$ and ${\lim }_{h\to 0}\frac{\cos h-1}{h}=0$. For $f(x)=\sin x$, applying the definition gives: $$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{\sin (x+h)-\sin x}{h}$$ Using the sine addition identity $\sin (x+h)=\sin x\cos h+\cos x\sin h$, expand and simplify the numerator to get: $$f^{\prime }(x)=\underset{h\to 0}{\lim }\left(\sin x\cdot \frac{\cos h-1}{h}+\cos x\cdot \frac{\sin h}{h}\right)=\sin x\cdot 0+\cos x\cdot 1=\cos x$$ Repeating the same process for $f(x)=\cos x$ gives the result $\frac{d}{dx}[\cos x]=-\sin x$. These formulas only hold when $x$ is measured in radians, which is always the convention on the AP exam, so you never need to adjust for degrees. A useful memory trick is that derivatives of sine and cosine cycle every four steps: $\sin x\to \cos x\to -\sin x\to -\cos x\to \sin x$.

Worked Example

Find the slope of the line tangent to $f(x)=3\sin x-2\cos x$ at $x=\frac{\pi }{3}$. Leave your answer in exact form.

1. Apply the constant multiple and sum differentiation rules to the function term-by-term.
2. Use the derivative rules for sine and cosine: $f^{\prime }(x)=3\cdot \frac{d}{dx}[\sin x]-2\cdot \frac{d}{dx}[\cos x]=3\cos x-2(-\sin x)=3\cos x+2\sin x$.
3. Evaluate the derivative at $x=\frac{\pi }{3}$, using unit circle values: $\cos \left(\frac{\pi }{3}\right)=\frac{1}{2}$ and $\sin \left(\frac{\pi }{3}\right)=\frac{\sqrt{3}}{2}$.
4. Substitute to get the slope: $f^{\prime }\left(\frac{\pi }{3}\right)=3\left(\frac{1}{2}\right)+2\left(\frac{\sqrt{3}}{2}\right)=\frac{3}{2}+\sqrt{3}$.

Exam tip: AP exam questions almost always require exact form for answers involving radicals and $\pi$, so never convert to a decimal unless the question explicitly asks for a numerical approximation.

3. Derivative of the Natural Exponential Function $e^x$

The natural exponential function $f(x)=e^x$, where $e\approx 2.71828$ is Euler's number, has a unique and extremely useful property: it is its own derivative. To derive this from first principles, start with the limit definition: $$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{e^{x+h}-e^x}{h}=e^x\underset{h\to 0}{\lim }\frac{e^h-1}{h}$$ By definition, $e$ is the only positive base that satisfies ${\lim }_{h\to 0}\frac{e^h-1}{h}=1$, so this simplifies directly to $f^{\prime }(x)=e^x\cdot 1=e^x$. This result holds for all real values of $x$, and only applies to the natural exponential function with base $e$. For exponential functions with other bases, you will rewrite them using the natural exponential later in the course, but for this topic we focus exclusively on the base $e$ case. This derivative is the foundation for modeling all exponential growth and decay processes in real-world problems.

Worked Example

Given $g(x)=5e^x+4\sin x$, find $g^{\prime }(0)$, the value of the derivative at $x=0$.

1. Apply the sum and constant multiple rules to $g(x)$ before using our base derivative formulas.
2. Differentiate term-by-term: $g^{\prime }(x)=5\cdot \frac{d}{dx}[e^x]+4\cdot \frac{d}{dx}[\sin x]=5e^x+4\cos x$.
3. Evaluate at $x=0$, using the identities $e^0=1$ and $\cos 0=1$.
4. Substitute to get the result: $g^{\prime }(0)=5(1)+4(1)=9$.

Exam tip: Never misapply the power rule to $e^x$. If you find yourself writing $\frac{d}{dx}[e^x]=x{e}^{x-1}$, stop and remind yourself: the power rule only works when the exponent is a constant, not when the variable is in the exponent.

4. Derivative of the Natural Logarithm $\ln (x)$

$\ln (x)$ is the inverse function of the natural exponential $e^x$, defined only for $x>0$. To derive its derivative, we use inverse function differentiation: if $y=\ln x$, then by definition $e^y=x$. Differentiate both sides with respect to $x$: $$e^y\cdot \frac{dy}{dx}=1$$ Rearrange to solve for $\frac{dy}{dx}$: $\frac{dy}{dx}=\frac{1}{e^y}=\frac{1}{x}$, since $e^y=x$. So the derivative of $\ln x$ is the reciprocal function $\frac{1}{x}$, defined for all $x>0$. For the absolute value form $\ln |x|$, the derivative is also $\frac{1}{x}$ for all $x\ne 0$, which you will use extensively when integrating later in the course. It is important to remember the domain restriction: $\ln (x)$ is undefined for $x\le 0$, so its derivative is also undefined there.

Worked Example

Find the equation of the tangent line to $y=\ln (x)+2e^x$ at $x=1$. Write your final answer in slope-intercept form.

1. To find the tangent line, we need the point $(1,y(1))$ and the slope $y^{\prime }(1)$.
2. Calculate the point: $y(1)=\ln (1)+2e^1=0+2e=2e$, so the point is $(1,2e)$.
3. Differentiate term-by-term: $y^{\prime }(x)=\frac{d}{dx}[\ln x]+2\frac{d}{dx}[e^x]=\frac{1}{x}+2e^x$.
4. Calculate the slope at $x=1$: $y^{\prime }(1)=\frac{1}{1}+2e^1=1+2e$.
5. Use point-slope form and simplify to slope-intercept: $y-2e=(1+2e)(x-1)$, so $y=(1+2e)x-(1+2e)+2e=(1+2e)x-1$.

Exam tip: Always check your simplification of tangent line equations: constant terms often cancel out, and AP graders deduct points for incorrect final form even if you got the slope and point right.

5. Common Pitfalls (and how to avoid them)

• Wrong move: Writing $\frac{d}{dx}[\sin x]=-\cos x$ or $\frac{d}{dx}[\cos x]=\sin x$, swapping derivatives and signs. Why: Students confuse the order of trigonometric derivatives, mixing derivative rules with antiderivative rules. Correct move: Write the derivative cycle $\sin x\to \cos x\to -\sin x\to -\cos x$ at the top of your scratch paper at the start of the exam to confirm whenever you need it.
• Wrong move: Differentiating $e^x$ as $x{e}^{x-1}$ by applying the power rule. Why: Students see an exponent and automatically use the power rule, which only works for constant exponents. Correct move: Always check where the variable is: variable in the base = power rule; variable in the exponent, base $e$ = $\frac{d}{dx}[e^x]=e^x$.
• Wrong move: Writing $\frac{d}{dx}[\ln x]=\frac{1}{x}\ln x$, adding an extra $\ln$ term. Why: Students mix up the derivative of $\ln x$ with other rules, or confuse it with exponential derivatives. Correct move: Memorize that the derivative of $\ln x$ is just the reciprocal function $\frac{1}{x}$, no extra terms needed.
• Wrong move: Differentiating $\ln (-5)$ (a constant) and getting $-\frac{1}{5}$. Why: Students apply the $\frac{1}{x}$ rule to any expression with $\ln$, forgetting that $\ln$ of a constant is just a constant. Correct move: Always check if the term is a constant before differentiating; the derivative of any constant is 0.
• Wrong move: Writing $\frac{d}{dx}[e^x+\cos x]=e^x+\sin x$, keeping the wrong sign for cosine. Why: Students see the lack of a negative sign on $e^x$ or $\sin x$ and incorrectly carry the positive sign over to cosine. Correct move: Label the derivative of each term separately before combining to avoid sign confusion.
• Wrong move: Writing $\frac{d}{dx}[5\ln x]=5x$, turning the reciprocal into $x$. Why: Students confuse the derivative of $\ln x$ with the power rule for $x^{-1}$, or misread the problem. Correct move: After differentiating, double-check that $\frac{1}{x}$ does not get flipped to $x$ by mistake.

6. Practice Questions (AP Calculus BC Style)

Question 1 (Multiple Choice)

If $f(x)=2\cos x-4e^x+\ln x$, which of the following is equal to $f^{\prime }(x)$? A) $-2\sin x-4e^x+\frac{1}{x}$ B) $2\sin x-4e^x+\ln x$ C) $-2\sin x-4e^x+x$ D) $2\sin x+4e^x+\frac{1}{x}$

Worked Solution: Differentiate term by term using the rules from this chapter. First, the derivative of $2\cos x$ is $2(-\sin x)=-2\sin x$, which eliminates options B and D, which have the wrong sign and start with $2\sin x$. Next, the derivative of $-4e^x$ is $-4e^x$, which both remaining options A and C have correct. Finally, the derivative of $\ln x$ is $\frac{1}{x}$, which eliminates option C, which incorrectly writes $x$ instead of $\frac{1}{x}$. The correct answer is A.

Question 2 (Free Response)

Let $h(x)=3e^x-2\sin x+\ln x$, defined for $x>0$. (a) Find $h^{\prime }(x)$. (b) Find the slope of the tangent line to $h(x)$ at $x=\frac{\pi }{2}$. Leave your answer in exact form. (c) The function $h(x)$ has a horizontal tangent line at some $x=a>0$. State the value of $h^{\prime }(a)$, and explain what this means for the graph of $h(x)$ at $x=a$.

Worked Solution: (a) Differentiate term-by-term using the rules from this chapter: $$\frac{d}{dx}[3e^x]=3e^x,\frac{d}{dx}[-2\sin x]=-2\cos x,\frac{d}{dx}[\ln x]=\frac{1}{x}$$ So the final derivative is: $$h^{\prime }(x)=3e^x-2\cos x+\frac{1}{x}$$

(b) Evaluate $h^{\prime }(x)$ at $x=\frac{\pi }{2}$. Recall that $\cos \left(\frac{\pi }{2}\right)=0$, so substitute: $$h^{\prime }\left(\frac{\pi }{2}\right)=3e^{\frac{\pi }{2}}-2(0)+\frac{1}{\frac{\pi }{2}}=3e^{\frac{\pi }{2}}+\frac{2}{\pi }$$ This is the exact slope of the tangent line.

(c) A horizontal tangent line has a slope of 0, so $h^{\prime }(a)=0$. This means the instantaneous rate of change of $h(x)$ at $x=a$ is 0, and $x=a$ is a critical point of $h(x)$ (a candidate for a local maximum or local minimum, which will be confirmed later in the course when we do curve analysis).

Question 3 (Application / Real-World Style)

The population of bacteria in a petri dish $t$ hours after the start of an experiment is modeled by $P(t)=1000e^t+50\sin \left(\frac{\pi t}{12}\right)+200\ln (t+1)$, measured in bacteria cells, for $t\ge 0$. The sine term models daily fluctuations in temperature that affect population growth. Find the instantaneous rate of change of the population at $t=0$ hours, include units, and interpret your result.

Worked Solution: First, differentiate $P(t)$ term-by-term to get the rate of change function: $$P^{\prime }(t)=1000e^t+50\left(\frac{\pi }{12}\right)\cos \left(\frac{\pi t}{12}\right)+200\left(\frac{1}{t+1}\right)$$ Evaluate at $t=0$, using $e^0=1$, $\cos (0)=1$, and $\frac{1}{0+1}=1$: $$P^{\prime }(0)=1000(1)+\frac{25\pi }{6}(1)+200(1)=1200+\frac{25\pi }{6}\approx 1213$$ The units are bacteria cells per hour. At the moment the experiment starts, the bacteria population is increasing at a rate of approximately 1213 cells per hour.

7. Quick Reference Cheatsheet

8. What's Next

This chapter gives you the four core non-polynomial derivatives that all subsequent differentiation topics build on. Immediately after this, you will learn the product rule, quotient rule, and the chain rule, which you will use to combine these base derivatives to differentiate almost any function you encounter on the AP exam. Without memorizing these four base derivatives correctly, every more complex differentiation problem will be impossible to solve correctly, since all extended rules rely on these base results. This topic also feeds directly into integration, where you will reverse these derivative rules to find antiderivatives for trigonometric, exponential, and logarithmic functions.

Follow-on topics to study next: Product and Quotient Rules Derivatives of General Exponential and Logarithmic Functions The Chain Rule Antiderivatives