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

Lesson ~10 min read

In simple terms: In simple terms, this lesson gives you four new "shortcut" rules to find the rate of change (derivative) for the common functions sin(x), cos(x), e^x, and ln(x).

Why this matters

Imagine you're an audio engineer in a recording studio in Atlanta, mixing a new track for an artist. You're looking at the sound waves on your screen. The smooth, repeating wave of a pure tone looks a lot like a sine or cosine function. Now, what if you wanted to know the exact moment the sound is getting louder the fastest? That's not about the volume itself, but the rate of change of the volume.

Calculus gives us the tools to answer that question. You've already learned the Power Rule for functions like x^2. But what about these other essential functions that model everything from sound waves to population growth to radioactive decay?

In this lesson, we'll add four powerful new rules to your calculus toolbox for finding the derivatives of sin(x), cos(x), the natural exponential e^x, and the natural log ln(x).

Concept overview

flowchart TD
    A[Start: See a limit problem] --> B{Does it look like lim h->0 of [g(x)-g(a)]/h?};
    B -->|Yes| C[Identify the function f(x) and the point a];
    C --> D{Is f(x) one of our new functions? sin(x), cos(x), e^x, ln(x)?};
    D -->|Yes| E[Find the derivative function, f'(x)];
    E --> F[Calculate f'(a)];
    F --> G[Result is f'(a)];
    B -->|No| H[Use other limit strategies like algebra or L'Hopital's Rule];
    D -->|No| H;

Core explanation

Alright, let's get to it. So far, you've built a solid foundation with the Power Rule, Product Rule, and Quotient Rule. Think of those as your all-purpose wrenches and screwdrivers. Today, we're adding four specialist tools to your kit. These are for specific, common functions you'll see everywhere.

There's no deep, complex derivation you need to memorize for the exam. For now, I need you to commit these four rules to memory, just like you memorized your multiplication tables in elementary school.

The Four Essential Rules

Here they are. Let's treat them as facts of nature for now.

1
The Sine Rule
The derivative of sin(x) is cos(x). d/dx [sin(x)] = cos(x)
2
The Cosine Rule
The derivative of cos(x) is -sin(x). d/dx [cos(x)] = -sin(x) Pay close attention here! This is the first one with a surprise negative sign. It's one of the most common places to lose a point on the exam. The derivative of cosine is negative sine.
3
The Natural Exponential Rule
The derivative of e^x is... itself, e^x. d/dx [e^x] = e^x This is a truly special function! It's the only function (besides the zero function) that is its own derivative. Imagine a population of bacteria growing in a lab. If its growth rate is always equal to its current size, its growth would be modeled by e^x. It's a perfect model of unchecked, proportional growth.
4
The Natural Log Rule
The derivative of ln(x) is 1/x. d/dx [ln(x)] = 1/x This makes intuitive sense. When x is a large number (like 1000), the ln(x) graph is very flat—its slope is small. And 1/x (1/1000) is a very small number. When x is a tiny positive number (like 0.01), the ln(x) graph is incredibly steep—its slope is huge. And 1/x (1/0.01 = 100) is a very large number. The rule works.

These four rules now work with all your other rules. You can use them in the Product Rule, Quotient Rule, and with constants.

For example, if f(x) = 4e^x - 5sin(x), the derivative is simply: f'(x) = 4 * d/dx[e^x] - 5 * d/dx[sin(x)] f'(x) = 4e^x - 5cos(x)

You just apply the rules to each part.

The Derivative in Disguise (The Limit Connection)

Now for the clever part. The College Board loves to test whether you understand where derivatives come from. Remember the formal limit definition of a derivative?

f'(a) = lim (h→0) [f(a + h) - f(a)] / h

This formula represents the instantaneous rate of change of the function f(x) at the point x = a.

On the AP exam, you won't be asked to use this formula to prove the derivative of sin(x). Instead, you'll be given a problem that is the formula, and you have to work backward.

Let's say you see this question on a test:

Evaluate: lim (h→0) [e^(3+h) - e^3] / h

Your brain might want to plug in h=0, which gives (e^3 - e^3) / 0 = 0/0. That's an indeterminate form, which tells you there's more work to do. But don't reach for algebra! Instead, match it to the definition:

lim (h→0) [f(a + h) - f(a)] / h lim (h→0) [e^(3 + h) - e^3] / h

Look at how perfectly they line up!

By comparing the two, what is our function f(x)? It must be e^x.
What is our specific point a? It must be 3.

So, this entire complicated limit expression is just a fancy way of asking: "What is the derivative of f(x) = e^x evaluated at x = 3?"

Well, we know the derivative of e^x is just e^x. So, f'(x) = e^x. And f'(3) = e^3.

That's it. The answer is e^3. No complicated calculations needed. You just have to recognize the disguise.

This skill is about pattern recognition. When you see a limit with h→0 and a fraction that looks like [something - something] / h, your alarm bells should go off. Ask yourself: "Is this just the definition of the derivative in disguise?"

Worked examples

Let's walk through a couple of examples to make this concrete.

Example 1

Combining Rules

Problem: Let g(x) = 7cos(x) + 2e^x. Find the equation of the line tangent to g(x) at x = 0.

Solution Walkthrough:

1
Identify the Goal
We need the equation of a tangent line. For that, we always need two things: a point (x, y) and a slope m.
2
Find the Point
We're given the x-coordinate, x = 0. To find the y-coordinate, we just plug x=0 into the original function g(x). g(0) = 7cos(0) + 2e^0 Remember your unit circle and exponent rules! cos(0) = 1 and e^0 = 1. g(0) = 7(1) + 2(1) = 9. So, our point is (0, 9).
3
Find the Slope
The slope of the tangent line is the value of the derivative at that point. First, we need to find the derivative function, g'(x). We'll use our new rules. g'(x) = d/dx [7cos(x) + 2e^x] g'(x) = 7 * d/dx[cos(x)] + 2 * d/dx[e^x] g'(x) = 7(-sin(x)) + 2(e^x) g'(x) = -7sin(x) + 2e^x
4
Calculate the Slope at the Point
Now, plug x=0 into the derivative g'(x) to get the specific slope m. m = g'(0) = -7sin(0) + 2e^0 sin(0) = 0 and e^0 = 1. m = -7(0) + 2(1) = 2. The slope of our tangent line is 2.
5
Write the Equation
Now we use point-slope form, y - y₁ = m(x - x₁). y - 9 = 2(x - 0) y - 9 = 2x y = 2x + 9

Final Answer: The equation of the tangent line is y = 2x + 9. Where students go wrong: The most common error here is forgetting the negative sign when taking the derivative of cos(x). If you forget it, you'll get a slope of m=2, which in this case is correct, but that's just luck because sin(0)=0. If the point was x=π/2, the mistake would be obvious. Always be careful with that cosine rule!

Example 2

The Limit Disguise

Problem: Evaluate the limit: lim (h→0) [sin(π/2 + h) - 1] / h

Solution Walkthrough:

1
Analyze the Expression
We see lim (h→0) and the fraction structure. This should immediately make you think of the derivative definition: f'(a) = lim (h→0) [f(a + h) - f(a)] / h.
2
Match the Pattern
Let's line them up. [f(a + h) - f(a)] / h [sin(π/2 + h) - 1] / h The f(a+h) part seems to be sin(π/2 + h). This suggests our function is f(x) = sin(x) and our point is a = π/2.
3

Verify the f(a) part: If f(x) = sin(x) and a = π/2, then f(a) should be f(π/2) = sin(π/2). From the unit circle, we know sin(π/2) = 1. This matches the - 1 in the numerator perfectly!
4
Re-interpret the Problem
We've confirmed that this limit is just a fancy way of asking for the derivative of f(x) = sin(x) at the point x = π/2.
5
Solve the Simpler Problem
- First, find the derivative of f(x) = sin(x). Using our new rule, f'(x) = cos(x).
- Now, evaluate this derivative at a = π/2.
- f'(π/2) = cos(π/2) = 0.

Final Answer: The value of the limit is 0. Where students go wrong: Many students will see the -1 and not immediately connect it to sin(π/2). They get stuck because it doesn't look exactly like f(a). You have to be a detective—identify f and a from the first part of the expression, and then confirm that the second part matches.

Try it yourself

Ready to try a couple on your own? Don't just jump to the answer; talk yourself through the steps we just practiced.

Problem 1: A particle's position along the x-axis is given by p(t) = 10 - 3sin(t) + ln(t), where t > 0. Find the particle's velocity function, v(t). What is the particle's acceleration at time t = π?

Problem 2: Use the definition of the derivative to evaluate the following limit: lim (h→0) [cos(π + h) + 1] / h

TL;DR

In simple terms, this lesson gives you four new "shortcut" rules to find the rate of change (derivative) for the common functions sin(x), cos(x), e^x, and ln(x).

Key terms

Derivative of sin x Derivative of cos x Derivative of e^x Derivative of ln x AP Calculus Limit definition of derivative Differentiation rules Calculus help FUN-3.A LIM-3.A

You can now…

FUN-3.A: Calculate derivatives of familiar functions.
LIM-3.A: Interpret a limit as a definition of a derivative.

Essential knowledge (exam-tested)

FUN-3.A.4: Specific rules can be used to find the derivatives for sine, cosine, exponential, and logarithmic functions.
LIM-3.A.1: In some cases, recognizing an expression for the definition of the derivative of a function whose derivative is known offers a strategy for determining a limit.

Concept map

flowchart TD
    A[Start: See a limit problem] --> B{Does it look like lim h->0 of [g(x)-g(a)]/h?};
    B -->|Yes| C[Identify the function f(x) and the point a];
    C --> D{Is f(x) one of our new functions? sin(x), cos(x), e^x, ln(x)?};
    D -->|Yes| E[Find the derivative function, f'(x)];
    E --> F[Calculate f'(a)];
    F --> G[Result is f'(a)];
    B -->|No| H[Use other limit strategies like algebra or L'Hopital's Rule];
    D -->|No| H;

Read what Saavi narrates

Hey everyone, it's Saavi from Shrutam.

Imagine you're an audio engineer in a studio, looking at the smooth, repeating sound wave of a pure tone on your screen. That wave can be modeled perfectly by a sine or cosine function. And if you wanted to find the exact moment the sound is getting louder the fastest, you'd need to find the peak rate of change. That's a calculus problem!

Today, we're adding four new, powerful rules to your calculus toolbox. These are the derivatives for sine, cosine, the natural exponential e to the x, and the natural log of x. These are shortcuts you just have to memorize, but they unlock so much.

The derivative of sine of x... is cosine of x.
The derivative of cosine of x... is negative sine of x. Be careful with that negative sign, it trips a lot of people up.
The derivative of e to the x... is just e to the x. It's its own derivative, which is pretty amazing.
And the derivative of the natural log of x... is one over x.

Now, let's look at how the AP exam will test this in a clever way. You might see a problem that says: Evaluate the limit as h approaches zero, of, the fraction, sine of pi over 2 plus h, minus 1, all over h.

Now, your first instinct might be to panic. But pause. Remember the limit definition of a derivative? It looks just like this. The formula is the limit as h approaches zero of f of a plus h, minus f of a, all over h.

Let's match them up. Our function, f of x, must be sine of x. And our point, a, must be pi over 2. And what is sine of pi over 2? It's 1. So that minus 1 in the problem is just f of a.

The whole scary limit problem is just asking: "What's the derivative of sine of x, at the point x equals pi over 2?"

Well, we know the derivative of sine of x is cosine of x. And cosine of pi over 2 is zero. So the answer is just... zero. See? Once you recognize the pattern, it's simple.

One of the most common mistakes I see is students forgetting the negative sign on cosine's derivative. They'll say the derivative of cosine is sine. It's not! It's negative sine. Remember, the "co-" functions, like cosine, cotangent, and cosecant, all have negative derivatives. It's a good pattern to remember.

You've got this. Keep practicing recognizing that limit pattern, and these four rules will become second nature.

Back to AP Calculus AB Take a mock exam