What Is Differentiation?

Before you begin

You should be comfortable with functions, powers (indices), and the gradient of a straight line.

The intuition

A straight line has a constant gradient. A curve has a gradient that changes from point to point. The derivative $f'(x)$ tells you the gradient of the curve at any value of $x$ — the gradient of the tangent line touching the curve at that point.

Definitions

• Derivative: $f'(x)$ or $\frac{dy}{dx}$, the function that gives the gradient of $y = f(x)$ at each point.
• Differentiation: the process of finding the derivative.
• Tangent: a straight line that touches a curve at a single point with the same gradient as the curve there.

The basic rule

For $y = x^n$, where $n$ is a constant: $\frac{dy}{dx} = n x^{n-1}$ For a sum or difference, differentiate each term separately. Constant terms differentiate to $0$.

Step-by-step method

1. Write each term as a power of $x$ (for example, $\frac{1}{x} = x^{-1}$ and $\sqrt{x} = x^{1/2}$).
2. Apply $nx^{n-1}$ to each term.
3. Simplify the indices.
4. Add the constant rule: the derivative of a constant is $0$.

Worked examples

Foundation example

Differentiate $y = x^5$. $\frac{dy}{dx} = 5x^{5-1} = 5x^4$

Developing example

Differentiate $y = 4x^3 - 7x^2 + 3x - 9$. Differentiate each term: $\frac{dy}{dx} = 12x^2 - 14x + 3$ (The constant $-9$ differentiates to $0$.)

Advanced example

Differentiate $y = \frac{1}{x^2} + \sqrt{x}$ with positive indices. Rewrite: $y = x^{-2} + x^{1/2}$. $\frac{dy}{dx} = -2x^{-3} + \frac{1}{2}x^{-1/2} = -\frac{2}{x^3} + \frac{1}{2\sqrt{x}}$

Common mistakes

• Forgetting to reduce the index: $\frac{d}{dx}(x^5) = 5x^4$, not $5x^5$.
• Ignoring constants: the derivative of $+7$ is $0$.
• Rewriting powers incorrectly: $\sqrt{x} = x^{1/2}$, not $x^2$.

Concept checklist

• I can differentiate $x^n$ using the power rule.
• I can differentiate sums and constant terms.
• I can rewrite roots and reciprocals as powers of $x$ first.

Practice questions

1. Differentiate $y = x^7$.
2. Differentiate $y = 3x^4 - 5x^2 + 8$.
3. Differentiate $y = x^{-3}$.
4. Differentiate $y = \sqrt{x}$.
5. Find the gradient of $y = x^2$ at $x = 4$.
6. Differentiate $y = \frac{6}{x}$ with positive indices.

Challenge questions

1. Differentiate $y = \frac{3}{x^2} - 2\sqrt{x}$ with positive indices.
2. Find the value of $x$ where the gradient of $y = x^2 - 6x + 5$ is zero.

Frequently asked questions

What does the derivative tell you? It gives the gradient (rate of change) of the curve at any point. A positive derivative means the function is increasing; a negative derivative means it is decreasing.

Why do constants disappear? A constant does not change, so its rate of change is $0$.

Related concepts

• Differentiation from first principles
• The product, quotient and chain rules
• Tangents, normals and rates of change
• Maximum and minimum problems using differentiation