Derivatives of cos, sin, e^x, ln x — Derivatives of cos x, sin x, ex, and ln x - Calculus 1

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.

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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.

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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.

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.

Summary of the four essential derivative formulas

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

1
Step 1 — Identify Each Term

The 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.
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Step 2 — Differentiate 3 sin x

Using 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
3
Step 3 — Differentiate 2eˣ

Using d/dx [e^x] = e^x and the constant multiple rule: d/dx [2e^x] = 2e^x.

2eˣ
4
Step 4 — Differentiate −5 cos x

Using 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
5
Step 5 — Differentiate 4 ln x

Using d/dx [ln x] = 1/x: d/dx [4 ln x] = 4 × (1/x) = 4/x.

4/x
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Step 6 — Combine All Terms

Adding the derivatives of all four terms together gives the final answer.

f′(x) = 3 cos x + 2eˣ + 5 sin x + 4/x

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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.

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.

Five common errors and their corrections

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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.

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′

How the basic rules extend with the chain rule

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.

Question 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.

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At x = 0, sin x is increasing (its slope is positive), and cos 0 = 1, which is positive — no negative sign needed. However, at x = 0, cos x is at its peak and about to decrease. Just after x = 0, the slope of cos x is negative, and −sin x gives a negative value as x moves past zero. The negative sign in d/dx [cos x] = −sin x reflects the fact that as you move rightward from x = 0, the cosine curve falls while the sine curve rises. Graphically, the slopes of cos x are always the mirror image (negation) of the values of sin x.

Question 2: BASIC CALCULATION

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

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Apply the constant multiple rule and sum rule to each term separately. The derivative of 7 sin x is 7 cos x (since d/dx [sin x] = cos x). The derivative of −4e^x is −4e^x (since d/dx [e^x] = e^x). Combining these gives f′(x) = 7 cos x − 4e^x.

Question 3: INTERMEDIATE

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

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First, find the y-coordinate: g(1) = e¹ + ln 1 = e + 0 = e ≈ 2.718. Next, find the derivative: g′(x) = e^x + 1/x. Evaluate it at x = 1: g′(1) = e¹ + 1/1 = e + 1 ≈ 3.718. The tangent line passes through (1, e) with slope e + 1. Using point-slope form: y − e = (e + 1)(x − 1), which simplifies to y = (e + 1)x − 1. This tangent line approximates g(x) near x = 1.

Question 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.

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Velocity is the derivative of position, so we need h′(t). Using our derivative rules: h′(t) = 2 cos t − (−sin t) = 2 cos t + sin t. Now substitute t = π/3: h′(π/3) = 2 cos(π/3) + sin(π/3) = 2 × (1/2) + (√3/2) = 1 + √3/2 ≈ 1 + 0.866 = 1.866 m/s. The positive value means the buoy is moving upward at approximately 1.87 meters per second at t = π/3 seconds.

Question 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.

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Starting with f(x) = sin x, the successive derivatives are: f′(x) = cos x, f″(x) = −sin x, f‴(x) = −cos x, and f⁽⁴⁾(x) = sin x. After four differentiations, you return to the original function, creating a cycle of period 4: sin x → cos x → −sin x → −cos x → sin x → … This pattern occurs because sine and cosine are related by a phase shift of π/2 radians, and each differentiation shifts the function by π/2. Four shifts of π/2 equal a full cycle of 2π, returning you to the start. This cyclic behavior is unique to sinusoidal functions and has important applications in differential equations and signal processing.

SUMMARY

Lesson Summary

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.