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

Introduction: Why these derivatives matter

students, this lesson covers four of the most important functions in AP Calculus AB: $\sin x$, $\cos x$, $e^x$, and $\ln x$. These functions appear in models for motion, growth, waves, sound, cooling, population change, and many other real-world situations 🌊📈. Knowing their derivatives lets you find rates of change quickly and accurately.

By the end of this lesson, you should be able to:

• State and use the derivatives of $\sin x$, $\cos x$, $e^x$, and $\ln x$
• Explain why these rules fit into the bigger idea of differentiation
• Apply derivative rules to solve problems in AP Calculus AB
• Connect these functions to continuity, differentiability, and the meaning of slope

A derivative tells you how fast a function is changing at a point. If a position function tells where something is, its derivative tells velocity. If a function models temperature or money, its derivative tells how fast those values are changing. The four functions in this lesson are especially important because their derivatives have simple patterns and show up constantly in calculus.

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

The trigonometric functions $\sin x$ and $\cos x$ are fundamental because they describe repeating motion, such as a pendulum swinging or a wheel turning 🔄. Their derivatives are based on how the graph is changing at each point.

The key derivative rules are:

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

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

These formulas are best remembered together because they are linked. The derivative of $\sin x$ is $\cos x$, and the derivative of $\cos x$ is negative $\sin x$.

Why this makes sense

Think about the graph of $\sin x$. At $x=0$, the graph crosses the $x$-axis and rises upward. The slope there is positive, and $\cos 0=1$, which matches that positive slope. At $x=\frac{\pi}{2}$, the graph of $\sin x$ reaches a maximum, so the slope is $0$, and $\cos\left(\frac{\pi}{2}\right)=0$.

For $\cos x$, the graph starts at $1$ when $x=0$ and begins decreasing. That means the slope is negative. Since $-\sin 0=0$, the slope at the start is $0$, which matches the flat top of the cosine graph at $x=0$. As $x$ increases, the derivative becomes negative, matching the downward motion of the graph.

Example

If $f(x)=3\sin x$, then using the constant multiple rule,

$$f'(x)=3\cos x$$

If $g(x)=\cos x-2\sin x$, then

$$g'(x)=-\sin x-2\cos x$$

This shows how basic trig derivatives combine with other derivative rules.

The derivative of $e^x$

The exponential function $e^x$ is one of the most important functions in calculus. It is special because its derivative is itself:

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

This means the rate of change of $e^x$ at any point equals the function value at that point. In real life, this behavior is useful for continuous growth and decay models, such as compound interest, bacterial growth, and radioactive decay.

Why this is special

Most functions change shape when differentiated. For example, the derivative of $x^2$ is $2x$, which is a different function. But $e^x$ keeps the same form. That makes it uniquely important.

If a quantity grows proportionally to its current amount, then an exponential model often appears. For example, if the population of a city grows faster when the population is larger, $e^x$ helps describe that pattern.

Example

If $h(x)=5e^x$, then

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

If $p(x)=e^x+\sin x$, then

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

Here, the derivative of each term is found separately using the sum rule.

The derivative of $\ln x$

The natural logarithm $\ln x$ is the inverse of $e^x$. Since $e^x$ and $\ln x$ are inverse functions, their derivatives are closely connected. The derivative rule is:

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

The domain matters here. The function $\ln x$ is only defined for positive $x$, so its derivative is also only valid for $x>0$.

Why this derivative matters

The function $\ln x$ appears in problems involving growth rates, pH levels, information theory, and situations where changes are relative rather than absolute. Since $\frac{1}{x}$ gets smaller as $x$ gets larger, the slope of $\ln x$ decreases as $x$ increases. This matches the graph of $\ln x$, which rises quickly at first and then slowly flattens out.

Example

If $q(x)=\ln x$, then

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

If $r(x)=2\ln x+7$, then

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

If $s(x)=\ln x+e^x$, then

$$s'(x)=\frac{1}{x}+e^x$$

students, notice that the derivative of $\ln x$ is simpler than the function itself. This is useful when analyzing rates of change in models that use logarithms.

How these derivatives fit into the bigger picture

These four derivative rules are not isolated facts. They connect to the broader AP Calculus AB ideas of differentiability, continuity, and core differentiation rules.

A function must be continuous at a point to be differentiable there, but continuity alone does not guarantee differentiability. The functions $\sin x$, $\cos x$, $e^x$, and $\ln x$ are differentiable everywhere in their domains. That means their graphs have no sharp corners or vertical tangents where the derivative exists.

These derivatives also combine with the main differentiation rules:

• Constant rule
• Constant multiple rule
• Sum and difference rules
• Product rule
• Quotient rule
• Chain rule

For example, if $y=\sin(3x)$, you use the chain rule:

$$\frac{dy}{dx}=\cos(3x)\cdot 3=3\cos(3x)$$

If $y=\ln(2x)$, then

$$\frac{dy}{dx}=\frac{1}{2x}\cdot 2=\frac{1}{x}$$

If $y=e^{x^2}$, then

$$\frac{dy}{dx}=e^{x^2}\cdot 2x$$

These examples show that knowing the basic derivatives helps you handle more advanced expressions.

Common mistakes and how to avoid them

A few errors appear often in calculus work:

1. Forgetting the negative sign in the derivative of $\cos x$

• Correct: $\frac{d}{dx}(\cos x)=-\sin x$
• Incorrect: $\frac{d}{dx}(\cos x)=\sin x$

1. Writing $\frac{d}{dx}(e^x)=x e^{x-1}$

• This is not correct. The derivative of $e^x$ is still $e^x$

1. Forgetting the domain of $\ln x$

• The derivative $\frac{1}{x}$ only applies for $x>0$

1. Mixing up $\sin x$ and $\cos x$

• A quick check: the derivative of $\sin x$ should match a positive slope near $x=0$

1. Ignoring the chain rule when there is more than just $x$ inside the function

• Example: $\frac{d}{dx}(\sin(5x))\neq\cos(5x)$
• Correct: $5\cos(5x)$

A strong habit is to slow down and check whether the inside function needs an extra factor. This prevents many mistakes.

Conclusion

students, the derivatives of $\sin x$, $\cos x$, $e^x$, and $\ln x$ are core tools in AP Calculus AB. They are:

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

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

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

$$\frac{d}{dx}(\ln x)=\frac{1}{x}$$

These formulas help you study slopes, motion, growth, and change in many settings. They also work together with the main derivative rules to handle more complicated functions. Understanding these four derivatives gives you a strong foundation for the rest of calculus and for the AP exam 💡.

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 $x>0$
• $\sin x$ and $\cos x$ are differentiable for all real $x$
• $e^x$ is differentiable for all real $x$
• $\ln x$ is differentiable only where $x>0$
• Differentiability requires continuity, but continuity does not always guarantee differentiability
• Use the chain rule for expressions like $\sin(3x)$, $e^{x^2}$, and $\ln(2x)$
• Always check signs, especially for $\cos x$
• These derivatives are essential for AP Calculus AB problem-solving and modeling