Differentiation — Rules and Examples

Differentiation means finding a derivative. A derivative tells you how fast a function is changing at a point, so in calculus it is used for slope and rate of change questions.

The fastest way to choose the right rule is to look at the structure first. Is the expression a power like $x^5$, a sum like $x^3 + 2x$, a product like $x^2 e^x$, or a function inside another function like $(3x+1)^4$? The derivative rule depends on that structure.

Which Differentiation Rule Should You Use?

Start with the outermost shape of the expression.

• If the expression is a single power of $x$, use the power rule.
• If terms are added or subtracted, differentiate term by term.
• If two changing expressions are multiplied, use the product rule.
• If one changing expression is divided by another, use the quotient rule.
• If one function sits inside another, use the chain rule.

Many exercises use more than one rule. In that case, pick the rule that matches the outside structure first.

Main Differentiation Rules

Constant Rule

If $c$ is a constant, then:

$$\frac{d}{dx}(c) = 0$$

A fixed number does not change when $x$ changes.

Power Rule

If $n$ is a real number, then:

$$\frac{d}{dx}(x^n) = nx^{n-1}$$

Example: $\frac{d}{dx}(x^4) = 4x^3$.

Constant Multiple Rule

If $c$ is constant and $f$ is differentiable, then:

$$\frac{d}{dx}[cf(x)] = c f'(x)$$

The constant stays in front.

Sum And Difference Rule

If $f$ and $g$ are differentiable, then:

$$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$$

Differentiate each term separately, then keep the same plus or minus sign.

Product Rule

If $f$ and $g$ are differentiable, then:

$$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$

Use this when both factors depend on $x$.

Quotient Rule

If $f$ and $g$ are differentiable and $g(x) \ne 0$, then:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$

The condition $g(x) \ne 0$ matters because division by zero is undefined.

Chain Rule

If $y = f(g(x))$, and both functions are differentiable where needed, then:

$$\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$$

Use this when one function is nested inside another.

Why Structure Matters In Differentiation

Differentiation rules are shortcuts for common expression shapes. If the expression is simple, one rule is often enough. If it is built from parts, you combine rules.

That is why students often make errors before they even start differentiating. The main skill is not algebra first. It is recognizing the outside structure before you compute anything.

Differentiation Example: Product Rule And Chain Rule Together

Find the derivative of:

$$y = x^2(3x+1)^4$$

The outside structure is a product, so start with the product rule. Let:

$$f(x) = x^2 \quad \text{and} \quad g(x) = (3x+1)^4$$

Then:

$$y' = f'(x)g(x) + f(x)g'(x)$$

Differentiate the first factor:

$$f'(x) = 2x$$

Now differentiate $g(x) = (3x+1)^4$. This needs the chain rule because the inner expression is $3x+1$, not just $x$:

$$g'(x) = 4(3x+1)^3 \cdot 3 = 12(3x+1)^3$$

Substitute both pieces:

$$y' = 2x(3x+1)^4 + x^2 \cdot 12(3x+1)^3$$

This is already a correct derivative. If you want a factored form:

$$y' = 2x(3x+1)^3[(3x+1) + 6x]$$ $$y' = 2x(3x+1)^3(9x+1)$$

The important step is not the factoring. It is noticing that the whole expression is a product, while one factor also requires the chain rule.

Common Differentiation Mistakes

1. Using the power rule on the whole expression when the function is actually a product or quotient.
2. Forgetting the inner derivative in the chain rule. For $(3x+1)^4$, the full derivative is $4(3x+1)^3 \cdot 3$.
3. Differentiating a product by multiplying the derivatives. In general, $[f(x)g(x)]' \ne f'(x)g'(x)$.
4. Losing track of conditions. The quotient rule requires the denominator to be nonzero.

When Differentiation Rules Are Used

Differentiation rules appear anywhere one quantity changes with respect to another. In calculus, they are used for tangent slopes, optimization, and curve sketching.

In physics, derivatives describe quantities such as velocity and acceleration. In economics or engineering, they are used when you need a marginal change or a rate of change.

Try A Similar Differentiation Problem

Differentiate $y = (x^3 + 1)(2x - 5)^2$ and decide which rule applies first. If your answer is missing two product-rule terms or the inner derivative from $(2x - 5)^2$, go back and check the outside structure before simplifying.