Derivatives of $\cos x$, $\sin x$, $e^x$, and $\ln x$

students, this lesson covers four of the most important derivative formulas in AP Calculus BC 🌟. These functions appear everywhere in science, engineering, economics, and data modeling, so learning their derivatives gives you powerful tools for describing change. By the end of this lesson, you should be able to explain where these derivative rules come from, use them correctly, and connect them to the larger ideas of differentiation.

What you should learn

In this lesson, you will:

Explain the derivative rules for $\sin x$, $\cos x$, $e^x$, and $\ln x$
Use those rules to find slopes and rates of change
Connect these derivatives to continuity and differentiability
Recognize why these functions are especially important in calculus and real-world modeling

A derivative tells us how fast a function is changing at a point. For example, if a car’s position is modeled by a function, its derivative gives velocity. If the temperature changes over time, a derivative tells how quickly it is rising or falling. The functions in this lesson are common because they model repeating patterns, growth, and logarithmic change.

The derivatives of $\sin x$ and $\cos x$

The trigonometric functions $\sin x$ and $\cos x$ are central in calculus because they describe periodic behavior, such as sound waves, seasonal temperatures, and rotating motion. Their derivatives are closely linked:

$\frac{d}{dx}(\sin x)=\cos x$

$\frac{d}{dx}(\cos x)=-\sin x$

These formulas are usually memorized, but they are also connected through the limit definition of the derivative. For example, the derivative of $\sin x$ comes from the limit

$\lim_{h\to 0}\frac{\sin(x+h)-\sin x}{h}$

Using trig identities and the special limits $\lim_{h\to 0}\frac{\sin h}{h}=1$ and $\lim_{h\to 0}\frac{1-\cos h}{h}=0$, calculus shows that the result is $\cos x$.

A helpful way to remember these rules is to think of the unit circle. As the angle increases, $\sin x$ and $\cos x$ keep cycling between $-1$ and $1$. Their rates of change also cycle, which is why the derivative of $\sin x$ becomes $\cos x$, and the derivative of $\cos x$ becomes $-\sin x$.

Example 1

Find the derivative of $f(x)=3\sin x-2\cos x$.

Using linearity of derivatives,

$f'(x)=3\cos x-2(-\sin x)=3\cos x+2\sin x$

This is a great example of how derivative rules combine with constants and sums.

Example 2

Suppose $g(x)=\sin x+\cos x$. Then

$g'(x)=\cos x-\sin x$

If you evaluate this at $x=0$, you get

$g'(0)=\cos 0-\sin 0=1-0=1$

So the slope of the tangent line at $x=0$ is $1$.

The derivative of $e^x$

The function $e^x$ is special because it is its own derivative:

$\frac{d}{dx}(e^x)=e^x$

This makes $e^x$ the natural exponential function. It appears in population growth, radioactive decay, compound interest, and many other processes where change is proportional to the amount present.

Why is this true? One reason comes from the limit definition:

$\frac{d}{dx}(e^x)=\lim_{h\to 0}\frac{e^{x+h}-e^x}{h}$

Factoring out $e^x$ gives

$\lim_{h\to 0}e^x\frac{e^h-1}{h}$

Since $e^x$ does not depend on $h$, it stays outside the limit. The key result is that

$\lim_{h\to 0}\frac{e^h-1}{h}=1$

so the derivative is $e^x$.

Example 3

Find the derivative of $h(x)=5e^x$.

Using the constant multiple rule,

$h'(x)=5e^x$

Example 4

Find the derivative of $p(x)=e^x+\sin x$.

Then

$p'(x)=e^x+\cos x$

This mixes exponential growth with oscillation, which is useful in modeling wave behavior with increasing amplitude or other combined effects.

The derivative of $\ln x$

The natural logarithm is defined only for $x>0$, and it has a derivative that is simple and useful:

$\frac{d}{dx}(\ln x)=\frac{1}{x}, \quad x>0$

This means the rate of change of $\ln x$ gets smaller as $x$ gets larger. That fits the graph of $\ln x$, which increases slowly and keeps flattening out.

A common way to justify this derivative uses the inverse function relationship between $\ln x$ and $e^x$. Since $\ln x$ is the inverse of $e^x$, their derivatives are connected. If $y=\ln x$, then $x=e^y$. Differentiating implicitly gives

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

Because $e^y=x$, this becomes

$\frac{dy}{dx}=\frac{1}{x}$

So the derivative of $\ln x$ is $\frac{1}{x}$.

Example 5

Find the derivative of $q(x)=\ln x+4$.

The constant disappears, so

$q'(x)=\frac{1}{x}$

Example 6

Find the derivative of $r(x)=2\ln x-\cos x$.

Using rules for sums and constant multiples,

$r'(x)=\frac{2}{x}+\sin x$

Notice that the derivative of $-\cos x$ is $\sin x$ because $\frac{d}{dx}(\cos x)=-\sin x$.

Differentiability and continuity

A function must be continuous at a point to be differentiable there, but continuity alone does not guarantee differentiability. The functions in this lesson are good examples of smooth behavior.

$\sin x$ is continuous for all real numbers and differentiable for all real numbers.
$\cos x$ is continuous for all real numbers and differentiable for all real numbers.
$e^x$ is continuous for all real numbers and differentiable for all real numbers.
$\ln x$ is continuous and differentiable only for $x>0$.

This matters because a derivative describes a tangent slope, and a function needs to be smooth enough for that slope to exist. For example, a function with a corner or jump would not be differentiable there. These four functions do not have corners or cusps in their domains, so their derivatives exist throughout their allowed inputs.

Example 7

Is $f(x)=\ln x$ differentiable at $x=-1$?

No, because $\ln x$ is not defined for negative $x$. Since the function does not exist there, it cannot be differentiable there.

Example 8

Is $g(x)=e^x$ differentiable at $x=2$?

Yes. Since $e^x$ is differentiable everywhere, its derivative exists at $x=2$ and equals $e^2$.

How these derivative rules work together

AP Calculus BC often asks you to combine these basic derivatives with the product rule, quotient rule, and chain rule. That is why it is important to know the core rules exactly.

For example, if

$F(x)=e^x\sin x$

then the product rule gives

$F'(x)=e^x\sin x+e^x\cos x$

$G(x)=\ln(3x)$

then the chain rule gives

$G'(x)=\frac{1}{3x}\cdot 3=\frac{1}{x}$

These rules often work together. Knowing the basic derivatives of $\sin x$, $\cos x$, $e^x$, and $\ln x$ helps you solve more complex problems quickly and accurately.

Conclusion

students, the derivatives of $\sin x$, $\cos x$, $e^x$, and $\ln x$ are among the most important formulas in calculus 📘. They describe how common functions change, help model real-world systems, and form the foundation for more advanced differentiation skills. Remember that $\sin x$ and $\cos x$ create a linked pair, $e^x$ is its own derivative, and $\ln x$ differentiates to $\frac{1}{x}$. These rules also show the close connection between differentiability and continuity, since each function is smooth throughout its domain.

Study Notes

$\frac{d}{dx}(\sin x)=\cos x$
$\frac{d}{dx}(\cos x)=-\sin x$
$\frac{d}{dx}(e^x)=e^x$
$\frac{d}{dx}(\ln x)=\frac{1}{x}$ for $x>0$
Differentiability implies continuity, but continuity does not always imply differentiability
$\sin x$, $\cos x$, and $e^x$ are differentiable for all real numbers
$\ln x$ is differentiable only when $x>0$
Use the sum rule, constant multiple rule, product rule, quotient rule, and chain rule together with these basic derivatives
These formulas are essential for AP Calculus BC problem solving and modeling change