Differentiation Rules and the Chain Rule

The limit definition of the derivative is correct but slow: each new function needs its own difference-quotient computation. A handful of rules, each proved once from the definition, replace that work. The power, constant-multiple, sum, product, and quotient rules together differentiate any polynomial or rational function by inspection; the chain rule extends them to composite functions.

The building-block rules

Start with the two simplest functions. A constant function $f(x)=c$ has a horizontal graph, so its slope is $0$ everywhere; the identity $f(x)=x$ has slope $1$.

Both fall straight out of the definition. The general pattern for powers is the one rule worth memorizing.

For a positive integer $n$, the factorization $x^n-a^n=(x-a)(x^{n-1}+x^{n-2}a+\cdots +a^{n-1})$ gives

because the limit has $n$ equal terms. The quotient rule below extends this to negative integers, and a proof for all real exponents waits until the logarithmic derivative supplies logarithmic differentiation. The rule holds throughout.

Example. The Power Rule reads off directly.

• $f(x)=x^6\Rightarrow f^{\prime }(x)=6x^5$.
• $y=x^{1000}\Rightarrow y^{\prime }=1000x^{999}$.
• $y=1/x=x^{-1}\Rightarrow y^{\prime }=-x^{-2}=-1/x^2$.
• $y=\sqrt[3]{x^{-2}}=x^{-2/3}\Rightarrow y^{\prime }=-\frac{2}{3}{x}^{-5/3}$.

Combining derivatives

New functions built from old ones by scaling, adding, or subtracting have derivatives built the same way.

Each follows from the matching limit law: a constant pulls out of a limit, and the limit of a sum is the sum of the limits. Together with the Power Rule they differentiate any polynomial term by term.

Example. For $y=x^8+12x^5-4x^4+10x^3-6x+5$,

Example. Horizontal tangents of $y=x^4-6x^2+4$ occur where the derivative vanishes:

giving the points $(0,4)$, $(\sqrt{3},-5)$, and $(-\sqrt{3},-5)$.

The Product and Quotient Rules

The derivative of a product is not the product of the derivatives. Taking $f(x)=x$ and $g(x)=x^2$ gives $f^{\prime }{g}^{\prime }=1\cdot 2x=2x$, whereas $(fg{)}^{\prime }=(x^3{)}^{\prime }=3x^2$. The correct formula was found by Leibniz.

The formula has an area reading. Take $f$ and $g$ as the side lengths of a rectangle of area $fg$. Increasing $x$ grows the sides by differentials $\mathbf{d}f$ and $\mathbf{d}g$, which adds two strips, $f\mathbf{d}g$ along the top and $g\mathbf{d}f$ along the side, plus a corner $\mathbf{d}f\mathbf{d}g$. The corner is second-order and drops out in the limit, leaving $\mathbf{d}(fg)=f\mathbf{d}g+g\mathbf{d}f$.

The rigorous proof subtracts and adds $f(x+h)g(x)$ inside the difference quotient to separate the two functions:

Since $f$ is continuous at $x$, ${\lim }_{h\to 0}f(x+h)=f(x)$, and the limit is $f{g}^{\prime }+g{f}^{\prime }$.

Example. If $h(x)=xg(x)$ with $g(3)=5$ and $g^{\prime }(3)=2$, then $h^{\prime }(x)=x{g}^{\prime }(x)+g(x)$, so $h^{\prime }(3)=3(2)+5=11$.

A quotient is handled by a companion rule.

The order matters: denominator times derivative of numerator, minus numerator times derivative of denominator, all over the denominator squared. Applied to $1/x^n$ it extends the Power Rule to negative integers:

Example. For $y=\frac{x^2+x-2}{x^3+6}$,

The Quotient Rule is not always the fastest route. To differentiate $F(x)=(3x^2+2\sqrt{x})/x$, divide first: $F(x)=3x+2x^{-1/2}$, then apply the Power Rule.

The rules proved so far are collected below.

Any polynomial is differentiable on all of $\mathbb{R}$, and any rational function is differentiable on its domain.

Derivatives of the trigonometric functions

Throughout, $\sin x$ means the sine of the angle whose radian measure is $x$; the formulas below hold only in radians. Sketching the slope of $\sin x$ suggests that its derivative is $\cos x$, and the definition confirms it. Using the addition formula $\sin (x+h)=\sin x\cos h+\cos x\sin h$,

The evaluation reduces to two limits. One is proved by a squeeze between a sector and its bounding triangles; the other follows from it.

The first limit is proved by a squeeze on the unit circle. Take $0<\theta <\frac{\pi }{2}$ and place the angle at the centre $O$ of a circle of radius $1$, with $A$ on the initial side and $B$ on the terminal side. Drop $BC$ perpendicular to $OA$, and let the tangent at $A$ meet the line $OB$ produced at $E$. The triangle $OAB$, the circular sector $OAB$, and the triangle $OAE$ nest, so their areas increase in that order:

Dividing through by $\frac{1}{2}\sin \theta >0$ and taking reciprocals reverses the inequalities,

As $\theta \to 0^{+}$ the outer bound $\cos \theta \to 1$, so the squeeze forces $(\sin \theta )/\theta \to 1$; because the ratio is an even function, the two-sided limit is $1$ as well. The second limit follows by multiplying through by the conjugate $\cos \theta +1$:

Substituting these (with ${\lim }_{h\to 0}\sin x=\sin x$ and ${\lim }_{h\to 0}\cos x=\cos x$) leaves

The same method gives $\frac{\mathbf{d}}{\mathbf{d}x}\cos x=-\sin x$. Geometrically, the slope of the sine wave at each point equals the height of the cosine wave: the same curve shifted left by a quarter period.

The tangent, cotangent, secant, and cosecant derivatives follow from the Quotient Rule. For example,

The full table, valid only in radians, is worth memorizing; the minus signs attach to the three cofunctions.

Example. For $y=x^2\sin x$, the Product Rule gives $y^{\prime }=x^2\cos x+2x\sin x$.

The fundamental limit also resolves related trigonometric limits. To evaluate ${\lim }_{x\to 0}\frac{\sin 7x}{4x}$, arrange the argument to match:

The Chain Rule

The rules so far do not touch a function like $F(x)=\sqrt{x^2+1}$, which is a composite: with $u=g(x)=x^2+1$ and $y=f(u)=\sqrt{u}$, it is $F=f\circ g$. The chain rule differentiates the composition by multiplying the two rates of change.

The Leibniz form is a mnemonic for rates: if $u$ changes twice as fast as $x$ and $y$ changes three times as fast as $u$, then $y$ changes six times as fast as $x$. Multiplying $\mathbf{d}u$ formally cancels, though $\mathbf{d}u$ is not yet a genuine quantity to divide by; a careful proof handles the case where the inner increment $\Delta u$ can be zero.

The procedure is: differentiate the outer function at the inner function, then multiply by the derivative of the inner function.

Example. Composing the sine derivative with the chain rule, $\frac{\mathbf{d}}{\mathbf{d}x}\sin (x^2)=\cos (x^2)\cdot 2x=2x\cos (x^2)$, while $\frac{\mathbf{d}}{\mathbf{d}x}{\sin }^{2}x=2\sin x\cdot \cos x=\sin 2x$: the first squares inside the sine, the second squares outside it.

The Power Rule combined with the Chain Rule

The most common composite is a function raised to a power. With $u=g(x)$,

Example. $\frac{\mathbf{d}}{\mathbf{d}x}(x^3-1{)}^{100}=100(x^3-1{)}^{99}\cdot 3x^2=300x^2(x^3-1{)}^{99}$.

Longer compositions chain more links. For $y=f(u)$, $u=g(x)$, $x=h(t)$,

The Power Rule is a statement about how steepness scales with the exponent: higher powers sit flatter near the origin and climb faster past $x=1$, exactly as their derivatives $n{x}^{n-1}$ predict.