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

Welcome to the "Big Four": The Transcendental Derivatives

In your journey through AP Calculus, you have already learned the Power Rule (how to differentiate things like \( x^2 \) or \( 5x^3 \)). While the Power Rule is useful, the real world doesn't just move in straight lines or parabolas. Waves, sound, biological growth, and financial interest all follow different patterns.

In this chapter, we are going to learn the derivatives of the "Big Four" transcendental functions: sine, cosine, natural exponential, and natural logarithm. These formulas are the fundamental tools you will use for the rest of the year. Don't worry if they seem like a lot to memorize at first—with a few simple tricks, they will become second nature!

1. Trigonometric Derivatives: \( \sin x \) and \( \cos x \)

Trigonometric functions describe periodic behavior—anything that repeats in a cycle, like a swinging pendulum or the seasons of the year.

The Derivative of Sine

The derivative of the sine function is the cosine function:
\( \frac{d}{dx}(\sin x) = \cos x \)

The Derivative of Cosine

The derivative of the cosine function is the negative of the sine function:
\( \frac{d}{dx}(\cos x) = -\sin x \)

A Helpful Mnemonic: The "CO" Rule

One of the most common mistakes on the AP Exam is forgetting which trig derivative is negative. Here is a simple trick: Every "CO" function has a negative derivative. Since cosine starts with "co," its derivative (\( -\sin x \)) is negative. Later, when you learn the derivatives of cotangent and cosecant, you will see this rule still holds true!

Visualizing the Change

Think about the graph of \( y = \sin x \). At \( x = 0 \), the graph is climbing at its steepest point (a slope of \( 1 \)). Since \( \cos(0) = 1 \), the derivative formula matches the visual slope! At \( x = \pi/2 \), the sine graph "levels off" at a peak (a slope of \( 0 \)). Since \( \cos(\pi/2) = 0 \), the formula works there too.

Quick Takeaway:
\( \frac{d}{dx}(\sin x) = \cos x \)
\( \frac{d}{dx}(\cos x) = -\sin x \)

2. The Natural Exponential Function: \( e^x \)

The number \( e \) (approximately \( 2.718 \)) is a special constant in mathematics. It is often used to model population growth or radioactive decay.

The Derivative of \( e^x \)

The derivative of \( e^x \) is the easiest rule in all of calculus:
\( \frac{d}{dx}(e^x) = e^x \)

Yes, you read that correctly! The function \( e^x \) is its own derivative. This means that at any point on the graph of \( y = e^x \), the slope of the graph is exactly equal to the height of the graph.

The "Mirror" Analogy

Think of the derivative operator \( \frac{d}{dx} \) as a mirror for \( e^x \). When \( e^x \) looks into the mirror, it sees exactly itself. This unique property is why \( e \) is called the "natural" base.

Quick Takeaway:
If \( f(x) = e^x \), then \( f'(x) = e^x \). This is the only non-zero function that is its own derivative!

3. The Natural Logarithm: \( \ln x \)

The natural logarithm \( \ln x \) is the inverse of \( e^x \). It tells us the time needed to reach a certain level of growth.

The Derivative of \( \ln x \)

The derivative of the natural log function is a simple fraction:
\( \frac{d}{dx}(\ln x) = \frac{1}{x} \)

Important Note on Domain

Remember that you cannot take the logarithm of a negative number or zero. Therefore, this derivative rule only applies when \( x > 0 \). On the AP exam, you might see this written as \( \frac{d}{dx}(\ln|x|) = \frac{1}{x} \) to account for negative values of \( x \), but the basic rule remains the same: the derivative is the reciprocal of the input.

Why is this special?

Recall the Power Rule: \( \frac{d}{dx}(x^n) = nx^{n-1} \). If you tried to use the power rule to find an antiderivative for \( \frac{1}{x} \) (which is \( x^{-1} \)), the rule would break (you'd end up dividing by zero). The natural log function "fills the gap" in the power rule world.

Quick Takeaway:
The slope of the natural log curve at any point \( x \) is simply \( 1 \) divided by that \( x \) value.

4. Combining the Rules: Practice with Constants

Even though these rules look simple, they are often combined with the Constant Multiple Rule and the Sum/Difference Rules (covered in previous chapters). Let's look at how to handle them together.

Step-by-Step Example

Find the derivative of \( f(x) = 4\sin x + 2e^x - \ln x \).

1. Identify the parts: We have three separate terms.
2. Apply the rules to each term:
   - The derivative of \( 4\sin x \) is \( 4\cos x \).
   - The derivative of \( 2e^x \) is \( 2e^x \).
   - The derivative of \( -\ln x \) is \( -\frac{1}{x} \).
3. Combine them: \( f'(x) = 4\cos x + 2e^x - \frac{1}{x} \)

Common Pitfalls to Avoid

1. Mixing up Sine and Cosine Signs: Many students accidentally write \( \frac{d}{dx}(\cos x) = \sin x \). Always remember the "CO" rule: derivative of cosine is negative.

2. Treating \( e^x \) like the Power Rule: Don't bring the \( x \) down! A common mistake is writing \( \frac{d}{dx}(e^x) = xe^{x-1} \). This is wrong. The power rule only works when the base is the variable (like \( x^2 \)). When the exponent is the variable (like \( e^x \)), use the exponential rule.

3. Forgetting the Reciprocal: Some students confuse \( \frac{d}{dx}(\ln x) \) with \( e^x \). Remember: Logarithms turn into fractions (\( 1/x \)).

Quick Review Table

Use this table to quiz yourself before the next unit exam:

Function \( f(x) \)   →   Derivative \( f'(x) \)
\( \sin x \)   →   \( \cos x \)
\( \cos x \)   →   \( -\sin x \)
\( e^x \)   →   \( e^x \)
\( \ln x \)   →   \( \frac{1}{x} \)

Note: These rules are the foundation for the Product Rule, Quotient Rule, and Chain Rule, which you will study in the next chapters of this section. Mastering these four formulas now will make those complex techniques much easier to learn!