Derivative Rules Explained with Worked Examples

Why rules instead of limits

The derivative is defined as a limit, $f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}$, and the article what is a derivative works through that definition for simple functions. Doing so for every problem would be painful. Instead, mathematicians proved general rules once, and those rules now handle polynomials, roots, exponentials, logarithms, trigonometric functions and any combination of them.

Think of the rules in two groups: derivatives of the basic building blocks, and rules for combining blocks through sums, products, quotients and compositions.

The basic rules: constants, powers and sums

A constant never changes, so its rate of change is zero. The power rule works for any real exponent $n$, not just positive whole numbers, which makes it far more useful than it first appears. Constant multiples pass straight through, and the derivative of a sum is the sum of the derivatives.

Rewrite roots and fractions with $x$ in the denominator as powers before differentiating. Most power rule errors come from skipping this step.

Exponential, logarithmic and trigonometric functions

These derivatives are worth memorising, because every larger problem is built from them:

The function $e^x$ is its own derivative, which is why $e$ appears throughout calculus. The trigonometric rules require the angle in radians; in degrees, an extra factor of $\frac{\pi}{180}$ appears. The remaining trig derivatives are $\frac{d}{dx}\sec x = \sec x\tan x$, $\frac{d}{dx}\csc x = -\csc x\cot x$ and $\frac{d}{dx}\cot x = -\csc^2 x$, all of which follow from the quotient rule below.

The product rule

The derivative of a product is not the product of the derivatives. To see why, picture a rectangle with sides $f$ and $g$. When both sides grow slightly, the area gains a strip of size $f' \cdot g$ along one side and a strip of size $f \cdot g'$ along the other, plus a tiny corner that vanishes in the limit.

The quotient rule

Order matters here because of the subtraction. A common memory aid is low d-high minus high d-low, over low squared, where high is the numerator and low the denominator.

Applying it to $\tan x = \frac{\sin x}{\cos x}$ gives $\frac{\cos^2 x + \sin^2 x}{\cos^2 x} = \sec^2 x$, confirming the table above.

The chain rule

The chain rule handles compositions, functions inside functions. If $y = f(g(x))$, then:

In words: differentiate the outer function, leaving the inside alone, then multiply by the derivative of the inside. In Leibniz notation, with $u = g(x)$, it reads $\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$. Rates multiply: if $u$ changes 3 times as fast as $x$ and $y$ changes 2 times as fast as $u$, then $y$ changes 6 times as fast as $x$. Recognising the inner and outer functions is the skill practised in function composition.

Combining the rules

Real problems mix rules. Work from the outside in: decide what the last operation in the expression is, apply its rule, and then handle each piece.

When $y$ cannot be isolated at all, as in $x^2 + y^2 = 25$, the chain rule still works through implicit differentiation.

Common mistakes

• Forgetting the inner derivative in the chain rule, for example writing $\cos(x^2)$ for the derivative of $\sin(x^2)$.
• Multiplying derivatives for a product instead of using the product rule.
• Reversing the order in the numerator of the quotient rule, which flips the sign.
• Applying the power rule to $e^x$ or to $2^x$; those are exponential functions, not powers.
• Using degrees in trigonometric derivatives.

To check an answer, the derivative calculator shows which rule it applies at every step, and the graphing calculator lets you compare the slope of a curve with your derivative's values.

Inverse trigonometric functions

The inverse trigonometric functions have algebraic derivatives, which surprises many students the first time they see them. They come from differentiating the inverse relationship implicitly; for example, if $y = \arcsin x$ then $\sin y = x$, and differentiating both sides gives $\cos y \cdot y' = 1$. Since $\cos y = \sqrt{1 - x^2}$ on the range of arcsine, the result follows:

These two results matter more in integration than in differentiation, because they explain why integrals such as $\int \frac{dx}{1 + x^2}$ produce an arctangent.

Higher derivatives

Differentiating a derivative gives the second derivative, and the same rules apply each time. For $y = x^4$, the derivatives are $4x^3$, then $12x^2$, then $24x$, then 24, and every derivative after that is 0. A polynomial of degree $n$ always vanishes after $n + 1$ differentiations.

Sine and cosine cycle instead: the derivatives of $\sin x$ are $\cos x$, $-\sin x$, $-\cos x$ and then $\sin x$ again, repeating every four steps. The exponential $e^x$ never changes at all. These patterns are the starting point for building Taylor series, which approximate functions using all of their derivatives at a single point.

Practice problems with answers

Try each one before reading the answer. Decide first which rule is the outermost one.

1. $\frac{d}{dx}\left(2x^3 - \frac{1}{x}\right)$. Answer: $6x^2 + \frac{1}{x^2}$.
2. $\frac{d}{dx}\left(x^3 e^x\right)$. Answer: $x^2e^x(3 + x)$, by the product rule.
3. $\frac{d}{dx}\cos(5x + 1)$. Answer: $-5\sin(5x + 1)$, by the chain rule.
4. $\frac{d}{dx}\frac{\ln x}{x}$. Answer: $\frac{1 - \ln x}{x^2}$, by the quotient rule.
5. $\frac{d}{dx}\sqrt{1 + x^2}$. Answer: $\frac{x}{\sqrt{1 + x^2}}$, by the chain rule with the power $\frac{1}{2}$.

Rules in context: rates that depend on each other

The chain rule is what makes related rates problems possible. Suppose a circular ripple spreads on a pond so that its radius grows at 2 cm per second. The area is $A = \pi r^2$, and differentiating with respect to time gives $\frac{dA}{dt} = 2\pi r\frac{dr}{dt}$. When the radius is 10 cm, the area is growing at $2\pi(10)(2) = 40\pi \approx 125.7$ square centimetres per second.

The same idea answers questions about a ladder sliding down a wall, a balloon being inflated or the shadow of a person walking away from a street light. In each case you write an equation linking the quantities, differentiate every term with respect to time using the chain rule, and then substitute the values at the moment you care about. Substituting before differentiating is a classic mistake, because it turns changing quantities into constants.

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