2.7 Derivatives of cos x, sinx, e^x, and ln x

TLDR

The four key derivatives in AP Calculus topic 2.7 are: the derivative of sin x is cos x, the derivative of cos x is -sin x, the derivative of e^x is e^x, and the derivative of ln x is 1/x. You can combine these with the sum, difference, and constant multiple rules to differentiate expressions that mix trig, exponential, and logarithmic terms. You can also use these known derivatives backward to evaluate a limit that matches the definition of a derivative.

Why This Matters for the AP Calculus Exam

These four derivatives show up constantly across the AP Calculus exam, both on their own and inside bigger problems. On multiple-choice questions, you will need to apply them quickly and combine them with rules you already know. On free-response questions, they often appear as one step inside motion problems, tangent line problems, or rate-of-change setups, so recognizing them instantly saves time.

There is also a limit connection worth knowing. Sometimes a limit is secretly the definition of a derivative for a function whose derivative you already know. Spotting that lets you evaluate the limit without doing heavy algebra. For example, recognizing a difference quotient built from sin x means you can read off the answer as cos x at that point.

Key Takeaways

• The derivative of sin x is cos x, and the derivative of cos x is -sin x. Watch the negative sign on cosine.
• The derivative of e^x is e^x. It is its own derivative.
• The derivative of ln x is 1/x.
• Combine these with the constant multiple, sum, and difference rules to differentiate mixed expressions term by term.
• If a limit matches the form of a difference quotient for sin x, cos x, e^x, or ln x, you can recognize it as a derivative and skip the algebra.
• Keep notation clean: write each term's derivative separately before combining.

Derivatives of These Four Functions

Here is a quick table summarizing the rules.

Derivative of $\sin x$

The derivative of $\sin x$ is always $\cos x$. Here is an example:

$f(x) = 4\sin x +3x$

Differentiate $4\sin x$ and $3x$ separately.

Since the derivative of $\sin x$ is $\cos x$, the derivative of the first term is $4\cos x$. The derivative of $3x$ is $3$. So $f'(x) = 4\cos x+3$.

Derivative of $\cos x$

The derivative of $\cos x$ is always $-\sin x$. Here is an example:

$f(x) = 2\cos x +3$

Differentiate $2\cos x$ and $3$ separately.

Since the derivative of $\cos x$ is $-\sin x$, the derivative of the first term is $-2\sin x$. The derivative of $3$ is $0$, following the constant rule. So $f'(x) = -2\sin x$.

Derivative of $e^x$

The derivative of $e^x$ is simply $e^x$. It is its own derivative.

Here is an example:

$f(x) = e^x+3x^4$

The derivative of the first term is $e^x$. The derivative of the second term is $12x^3$, using the power rule. So $f'(x) = e^x + 12x^3$.

Derivative of $\ln x$

The derivative of $\ln x$ is $\frac{1}{x}$. Here is an example:

$f(x) = 5\ln x + 2x$

The derivative of the first term is $\frac{5}{x}$, since the derivative of $\ln x$ is $\frac{1}{x}$. The derivative of $2x$ is $2$, so $f'(x)=\frac{5}{x}+2$.

Using a Limit as a Derivative

Sometimes a limit is written in the exact form of the definition of a derivative. If the function inside is one whose derivative you know, you can recognize the limit as a derivative and read off the answer.

The two equivalent definition forms are:

$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \quad \text{and} \quad f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

For example, if you see a limit that matches $\lim_{h \to 0} \frac{\sin(a+h) - \sin a}{h}$, that is the definition of the derivative of $\sin x$ evaluated at $x = a$. Since the derivative of $\sin x$ is $\cos x$, the limit equals $\cos a$. The same idea works for $\cos x$, $e^x$, and $\ln x$: match the form, identify the function, then write its derivative at that point.

How to Use This on the AP Calculus Exam

MCQ

• Apply the four derivatives fast and combine them with the sum, difference, and constant multiple rules.
• Watch for the negative sign when differentiating cosine.
• Scan limit problems for difference quotient forms. If the inside function is sin x, cos x, e^x, or ln x, recognize the limit as a derivative instead of grinding through algebra.

Free Response

• These derivatives usually appear as one step inside a larger problem, such as finding a tangent line slope or an instantaneous rate of change.
• Show each term's derivative clearly before combining. Clean structure makes your work easy to follow and is important for clear exam work.
• Keep all notation, including parentheses and signs, accurate so a small slip does not change your answer.

Common Trap

• Do not write the derivative of cos x as +sin x. It is -sin x.
• Do not treat the derivative of e^x as x times something. It stays e^x.

Common Misconceptions

• The derivative of cos x is -sin x, not sin x. The negative sign is the most commonly dropped detail here.
• The derivative of e^x is e^x, not x·e^(x-1). The power rule does not apply to e^x because the base is constant and the variable is the exponent.
• The derivative of ln x is 1/x, not 1/x with an extra ln term. Keep it simple.
• A constant multiple stays in front: the derivative of 4 sin x is 4 cos x, not just cos x.
• Recognizing a limit as a derivative only works when the limit truly matches the difference quotient form for a known function. Check that the structure lines up before reading off the answer.

Related AP Calculus Guides

• Unit 2 Overview: Differentiation
• 2.1 Defining Average and Instantaneous Rates of Change at a Point
• 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
• 2.2 Defining the Derivative of a Function and Using Derivative Notation
• 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
• 2.3 Estimating Derivatives of a Function at a Point