How to Differentiate a Function- Calculus Tutorial

What Differentiation Actually Is

Differentiation finds the rate of change at any point on a curve. That's it. You take a function, apply the rules, and get another function that tells you the slope everywhere.

Why does this matter? Physics uses it for velocity and acceleration. Economics uses it for marginal cost. Engineering uses it for optimization. If you're taking calculus, you're going to need this.

The Basic Power Rule

This is where everyone starts. The power rule handles anything with an exponent.

Rule: If f(x) = xⁿ, then f'(x) = n·xⁿ⁻¹

Examples:

• f(x) = x⁵ → f'(x) = 5x⁴
• f(x) = x³ → f'(x) = 3x²
• f(x) = x² → f'(x) = 2x
• f(x) = x → f'(x) = 1
• f(x) = 1 → f'(x) = 0

The last two are worth remembering. Anything to the first power differentiates to 1. Constants differentiate to 0.

Constants and Coefficients

Constants multiply functions and stay put during differentiation.

Rule: d/dx [c·f(x)] = c·f'(x)

Example: If f(x) = 7x⁴, then f'(x) = 28x³

You multiply the coefficient by the exponent, then reduce the exponent by 1.

The Product Rule

When two functions multiply together, you can't just differentiate each part separately. That's a mistake beginners make constantly.

Rule: d/dx [f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)

Say it out loud: "First times derivative of second, plus second times derivative of first."

Example: Find the derivative of h(x) = x²·sin(x)

Let f(x) = x² and g(x) = sin(x)

f'(x) = 2x and g'(x) = cos(x)

h'(x) = 2x·sin(x) + x²·cos(x)

The Quotient Rule

Division requires the quotient rule. Yes, it's ugly. Yes, you need it.

Rule: d/dx [f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)] / [g(x)]²

mnemonic: "Low d-high minus high d-low, over low squared."

Example: Differentiate h(x) = x³/(x+1)

f(x) = x³, g(x) = x+1

f'(x) = 3x², g'(x) = 1

h'(x) = [3x²·(x+1) - x³·1] / (x+1)²

Simplify: h'(x) = [3x³ + 3x² - x³] / (x+1)² = [2x³ + 3x²] / (x+1)²

The Chain Rule

When functions nest inside other functions, you need the chain rule. Think of it as peeling layers.

Rule: d/dx [f(g(x))] = f'(g(x))·g'(x)

Take the derivative of the outer function, evaluated at the inner function, then multiply by the derivative of the inner function.

Example: Differentiate h(x) = (3x + 2)⁵

Outer function: u⁵ where u = 3x + 2

Derivative of outer: 5u⁴

Derivative of inner: 3

h'(x) = 5(3x + 2)⁴ · 3 = 15(3x + 2)⁴

Derivatives of Common Functions

You need to memorize these. They're the building blocks for everything else.

The exponential and logarithmic derivatives are especially important in higher math and applications.

Getting Started: Step-by-Step Process

Here's how to approach any differentiation problem:

Step 1: Identify the structure

Is it a single term? A product? A quotient? A composite function? The structure determines which rule you use.

Step 2: Apply the power rule first

Reduce all exponents by 1 and multiply by the original exponent.

Step 3: Handle coefficients

Multiply by any constant coefficient in front.

Step 4: Apply other rules as needed

Use product rule for products, quotient rule for divisions, chain rule for compositions.

Step 5: Simplify

Combine like terms. Factor if it makes the answer cleaner.

Worked example: Differentiate f(x) = 3x²(2x + 1)⁴

This combines product rule and chain rule.

Let u = 3x² and v = (2x + 1)⁴

u' = 6x

v' = 4(2x + 1)³ · 2 = 8(2x + 1)³

f'(x) = 6x·(2x + 1)⁴ + 3x²·8(2x + 1)³

f'(x) = 6x(2x + 1)⁴ + 24x²(2x + 1)³

You can factor out common terms if you want a cleaner answer.

Common Mistakes to Avoid

• Ignoring the chain rule — if you see parentheses or nested functions, you need it
• Using product rule on quotients — or vice versa. Know which situation you're in
• Forgetting negative exponents — they follow the same power rule
• Not simplifying — leaving unsimplified answers loses points
• Memorizing without understanding — knowing why the rules work helps you apply them correctly

Higher-Order Derivatives

You can differentiate a derivative. That's a second derivative. You can do it again for a third, fourth, and so on.

f''(x) = d/dx [f'(x)]

f'''(x) = d/dx [f''(x)]

Second derivatives show up in physics for acceleration (first derivative of position is velocity, second derivative is acceleration). They also tell you about concavity in calculus.

Implicit Differentiation

Sometimes y isn't isolated. You differentiate both sides with respect to x, treating y as a function of x.

Example: x² + y² = 25 (a circle)

Take d/dx of both sides:

2x + 2y·dy/dx = 0

Solve for dy/dx:

dy/dx = -x/y

This gives the slope of the tangent line at any point on the circle.