Derivatives of cos, sin, e^x, ln x — Derivatives of cos x,...

SECTION 1

Historical Context & Motivation

Calculus didn't appear overnight. For centuries, mathematicians struggled with a seemingly simple question: how do you measure the rate at which something changes at a single instant? The functions sin x, cos x, e^x, and ln x kept showing up in problems about planetary motion, population growth, and sound waves. Finding their derivatives — the formulas that describe how these functions change — became one of the great achievements of early calculus. Understanding the history helps you see why these four results matter so much.

~1670

Newton & Leibniz Invent Calculus

Isaac Newton and Gottfried Leibniz independently developed the foundations of calculus. Newton used fluxions (his word for derivatives) to study motion, while Leibniz created the dy/dx notation we still use today.

1748

Euler Formalizes eˣ and ln x

Leonhard Euler published his masterwork Introductio in analysin infinitorum, establishing e ≈ 2.71828 as a fundamental constant and showing that eˣ is its own derivative — a remarkable and unique property.

1821

Cauchy's Rigorous Limit Definition

Augustin-Louis Cauchy placed derivatives on solid logical ground by defining them through limits. This let mathematicians prove — not just guess — that the derivative of sin x is cos x.

1800s–Today

Applications Explode

These four derivative formulas became indispensable in physics, engineering, economics, and computer science. From modeling electromagnetic waves to pricing financial options, they appear everywhere.

So what exactly are the derivatives of sin x, cos x, e^x, and ln x, and why do they take the forms they do? That's the question this lesson answers, step by step.

SECTION 2

Core Principles & Definitions

Before memorizing formulas, you need to understand what a derivative actually tells you. The derivative of a function at any point gives the instantaneous rate of change — the slope of the tangent line at that point. For functions like sin x and e^x, these slopes follow elegant patterns that you can capture with simple formulas.

d/dx [sin x] = cos x

The rate at which sin x changes at any point equals the value of cos x at that same point. When the sine wave is climbing steepest (at x = 0), cos 0 = 1 — the maximum slope.

d/dx [cos x] = −sin x

The derivative of cosine is negative sine. The negative sign matters — it shows that when cos x is decreasing, the slope is negative, mirroring the behavior of −sin x.

d/dx [eˣ] = eˣ

The function eˣ is its own derivative. No matter where you are on the curve, the slope equals the function's value. This is the only function (up to constant multiples) with this property.

d/dx [ln x] = 1/x

The natural logarithm changes at a rate of 1/x. When x is small and positive, ln x changes quickly; as x grows large, the slope flattens toward zero.

✦ KEY TAKEAWAY

Think of each derivative formula as a speedometer reading. If your position on a road is described by sin x, then the speedometer always reads cos x. If your position is e^x, your speed is always exactly equal to how far you've traveled — the faster you go, the faster you accelerate. These four formulas are the speedometer calibrations for the most important functions in calculus.

SECTION 3

Visual Explanation — Functions and Their Derivatives

The best way to understand these derivative relationships is to see a function and its derivative plotted together. In the diagram below, you can observe how the slope of each original function (the solid curve) matches the value of its derivative (the dashed curve) at every point.

The solid blue curve is sin x. The dashed cyan curve is its derivative, cos x. Notice that wherever sin x reaches a peak or trough (slope = 0), the cos x curve crosses zero. Where sin x is climbing steepest (at x = 0), cos x equals 1.

This visual pattern reveals a deep truth: the value of the derivative at any x-coordinate equals the slope of the original function at that same x-coordinate. When you see the sine curve at a hilltop, its slope is zero, and sure enough, the cosine curve passes through zero right there. When the sine curve is falling steeply, the cosine curve dips to its most negative value.

SECTION 4

Mathematical Framework

Each of these four derivative formulas can be proven using the limit definition of the derivative: f′(x) = lim(h→0) [f(x + h) − f(x)] / h. Here we state each result precisely, define the variables, and note the domain.

DERIVATIVE OF SINE

d/dx [sin x] = cos x

Valid for all real numbers x, where x is measured in radians. This formula does not work if x is in degrees — always convert to radians first.

DERIVATIVE OF COSINE

d/dx [cos x] = −sin x

Valid for all real numbers x (in radians). The negative sign is the most common source of errors — always include it.

DERIVATIVE OF THE NATURAL EXPONENTIAL

d/dx [eˣ] = eˣ

Valid for all real numbers x. Here e ≈ 2.71828 is Euler's number. The function e^x is the only function (up to constant multiples) that equals its own derivative.

DERIVATIVE OF THE NATURAL LOGARITHM

d/dx [ln x] = 1/x

Valid for x > 0. Since ln x is only defined for positive inputs, its derivative 1/x shares this domain restriction. Note that 1/x is never zero, so the ln x curve never has a horizontal tangent.

⚠️ Radian Requirement

The derivatives of sin x and cos x only work when x is in radians. If you plug in degrees, you'll get the wrong slope. Always check your angle mode before differentiating trig functions.

SECTION 5

Detailed Breakdown — How the Four Derivatives Compare

Seeing all four derivative rules side by side makes it easier to spot patterns and avoid mix-ups. The table below summarizes each function, its derivative, domain, and a quick memory tip. The diagram that follows shows the exponential and logarithm pair together.

Summary of the four essential derivative formulas

Function f(x)	Derivative f′(x)	Domain	Memory Tip
sin x	cos x	All real numbers	"Sine becomes cosine" — they cycle together.
cos x	−sin x	All real numbers	"Cosine picks up a negative." Don't forget the minus!
eˣ	eˣ	All real numbers	"e to the x is its own twin." It never changes.
ln x	1/x	x > 0	"Log drops down to one over x." Reciprocal of the input.

The pink curve is e^x and the amber curve is ln x. They are reflections across the dashed line y = x because they are inverse functions. At (0, 1) on the e^x curve, the slope equals the value: both are 1. At (1, 0) on the ln x curve, the slope is 1/1 = 1 as well.

Notice a beautiful pattern: e^x and ln x are inverse functions (each undoes the other). Similarly, sin and cos are deeply linked — differentiating sin gives cos, and differentiating cos gives −sin. If you differentiate sin x four times, you cycle back to sin x. These cyclic and inverse relationships are why these four functions form a tightly connected family in calculus.

SECTION 6

Worked Example

Let's put these derivative rules to work with a function that combines all four types. We'll find the derivative of f(x) = 3 sin x + 2e^x − 5 cos x + 4 ln x. This will require the constant multiple rule and the sum/difference rule alongside our four formulas.

Find f′(x) for f(x) = 3 sin x + 2eˣ − 5 cos x + 4 ln x

Step 1 — Identify Each TermThe function has four terms: 3 sin x, 2e^x, −5 cos x, and 4 ln x. By the sum/difference rule, we differentiate each term separately and combine the results.

Step 2 — Differentiate 3 sin xUsing d/dx [sin x] = cos x and the constant multiple rule: d/dx [3 sin x] = 3 × cos x = 3 cos x.

3 cos x

Step 3 — Differentiate 2eˣUsing d/dx [e^x] = e^x and the constant multiple rule: d/dx [2e^x] = 2e^x.

2eˣ

Step 4 — Differentiate −5 cos xUsing d/dx [cos x] = −sin x: d/dx [−5 cos x] = −5 × (−sin x) = 5 sin x. The two negatives cancel, giving a positive result.

5 sin x

Step 5 — Differentiate 4 ln xUsing d/dx [ln x] = 1/x: d/dx [4 ln x] = 4 × (1/x) = 4/x.

4/x

Step 6 — Combine All TermsAdding the derivatives of all four terms together gives the final answer.

f′(x) = 3 cos x + 2eˣ + 5 sin x + 4/x

⚠️ Common Mistake Alert

In Step 4, many students forget that multiplying two negatives gives a positive. When you see −5 cos x, the derivative of cos x contributes another negative (−sin x), so −5 × (−sin x) = +5 sin x. Always watch your signs carefully!

SECTION 7

Common Errors & How to Avoid Them

These four derivative formulas are simple to state, but students frequently make errors when applying them. The table below lists the most common mistakes and provides strategies to avoid each one.

Five common errors and their corrections

Common Error	What Goes Wrong	How to Fix It
Missing the negative in d/dx [cos x]	Writing cos x → sin x instead of −sin x	Remember: "cosine carries a negative." Write −sin x immediately.
Using degrees instead of radians	The formulas give wrong slopes because they assume radian measure.	Always convert to radians. Set your calculator to radian mode.
Writing d/dx [eˣ] = xeˣ⁻¹	Confusing the exponential rule with the power rule d/dx [xⁿ] = nxⁿ⁻¹.	The power rule is for x raised to a constant. eˣ has the variable in the exponent — totally different.
Writing d/dx [ln x] = 1/(ln x)	Confusing 1/x with 1/(ln x).	The derivative of ln x is 1/x, not 1 divided by ln x. They look similar but are completely different expressions.
Forgetting domain of ln x	Evaluating d/dx [ln x] at x = 0 or x < 0.	ln x and 1/x share the domain x > 0. Neither exists at x = 0.

✦ KEY TAKEAWAY

Think of the power rule and the exponential rule like two different tools in a toolbox. The power rule (bring the exponent down, subtract one) is your wrench — it works when x is the base and the exponent is a constant. The exponential rule is a screwdriver — it's for when the base is a constant (like e) and x is in the exponent. Using the wrong tool gives you the wrong answer every time.

SECTION 8

Connection to the Chain Rule & Beyond

So far, we've differentiated simple versions of these functions — sin x, cos x, e^x, and ln x. But in real problems, you'll see things like sin(3x), e^2x, or ln(x²). To handle those, you'll need the chain rule, which multiplies your derivative by the derivative of the inner function. The table below previews how these formulas extend.

How the basic rules extend with the chain rule

Basic Rule (This Lesson)	With Chain Rule (Coming Soon)
d/dx [sin x] = cos x	d/dx [sin(u)] = cos(u) × u′
d/dx [cos x] = −sin x	d/dx [cos(u)] = −sin(u) × u′
d/dx [eˣ] = eˣ	d/dx [eᵘ] = eᵘ × u′
d/dx [ln x] = 1/x	d/dx [ln(u)] = (1/u) × u′

The formulas you learned today are the building blocks. Once you master the chain rule, you'll be able to differentiate nearly any function involving trig, exponentials, and logarithms. You'll also encounter implicit differentiation and logarithmic differentiation, both of which lean heavily on d/dx [ln x] = 1/x. Mastering these four basic rules now sets you up for success in every calculus topic ahead.

SECTION 9

Practice Problems

Test your understanding of the four derivative formulas with these five problems, arranged from conceptual to challenging. Try each one on paper before reading the answer.

PROBLEM 1 — CONCEPTUAL

Explain in your own words why the derivative of cos x has a negative sign, while the derivative of sin x does not.

PROBLEM 2 — BASIC CALCULATION

Find the derivative of f(x) = 7 sin x − 4e^x.

PROBLEM 3 — INTERMEDIATE

Find the equation of the tangent line to g(x) = e^x + ln x at the point where x = 1.

PROBLEM 4 — APPLIED

A buoy bobs up and down in the ocean. Its height above the water line (in meters) at time t seconds is h(t) = 2 sin t − cos t. Find the buoy's velocity at t = π/3 seconds.

PROBLEM 5 — CRITICAL THINKING

Suppose you differentiate sin x repeatedly. What is the fourth derivative of sin x? Describe the pattern you observe and explain why it occurs.

SUMMARY

Lesson Summary

This lesson covered the four fundamental derivative formulas that every calculus student must know. The derivative of sin x is cos x, and the derivative of cos x is −sin x — with the crucial negative sign that many students forget. The derivative of eˣ is eˣ itself, making it the only function that equals its own derivative. Finally, the derivative of ln x is 1/x, valid only for x > 0. All trig derivatives require radian measure.

These formulas combine with the constant multiple rule and sum/difference rule to let you differentiate any linear combination of these functions. Do not confuse the exponential rule with the power rule — the power rule handles xⁿ while the exponential rule handles eˣ. Looking ahead, the chain rule will extend these formulas to handle compositions like sin(3x) and e^2x. Master these four building blocks, and the rest of differential calculus becomes far more accessible.