Differentiation Rules

Definition

The core algebraic rules for computing derivatives: the power rule, linearity (sum and constant-multiple), the product rule, and the quotient rule.

Intuition

Each rule reduces differentiation of a compound expression to differentiation of its simpler parts, making it possible to differentiate any combination of elementary functions by breaking it down step by step.

Formal Description

Power Rule

For a positive integer $n$, applying the binomial expansion to the limit definition yields:

$$\frac{d}{dx}{x}^n=\underset{h\to 0}{\lim }\frac{(x+h{)}^n-x^n}{h}=\underset{h\to 0}{\lim }\frac{x^n+nh{x}^{n-1}+\cdots -x^n}{h}=n{x}^{n-1}.$$

Extension to all real exponents: For any real $p$ and $x>0$, write $x^p=\exp (p\ln x)$ and apply the chain rule:

$$\frac{d}{dx}{x}^p=\exp (p\ln x)\cdot \frac{p}{x}=x^p\cdot \frac{p}{x}=p{x}^{p-1}.$$ $$\boxed{\frac{d}{dx}{x}^n=n{x}^{n-1}}$$

Example — square root:

$$\frac{d}{dx}\sqrt{x}=\underset{h\to 0}{\lim }\frac{\sqrt{x+h}-\sqrt{x}}{h}=\underset{h\to 0}{\lim }\frac{1}{\sqrt{x+h}+\sqrt{x}}=\frac{1}{2\sqrt{x}}=\frac{1}{2}{x}^{-1/2}.$$

Sum and Constant-Multiple Rules

Differentiation is linear: the derivative of a sum (or difference) equals the sum (or difference) of the derivatives, and the derivative of a constant multiple equals that constant times the derivative.

Sum rule: For differentiable functions $f(x)$ and $g(x)$,

$$(f(x)+g(x){)}^{\prime }=\underset{h\to 0}{\lim }\frac{[f(x+h)+g(x+h)]-[f(x)+g(x)]}{h}=f^{\prime }(x)+g^{\prime }(x).$$

Constant-multiple rule:

$$(cf(x){)}^{\prime }=\underset{h\to 0}{\lim }\frac{cf(x+h)-cf(x)}{h}=c\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}=c{f}^{\prime }(x).$$ $$(f\pm g{)}^{\prime }=f^{\prime }\pm g^{\prime },(cf{)}^{\prime }=c{f}^{\prime }.$$

The derivative of any constant function is zero.

Product Rule

The product rule gives the derivative of a product of two differentiable functions. Starting from the limit definition, adding and subtracting $f(x)g(x+h)$ in the numerator:

$$\begin{array}{rl}[f(x)g(x){]}^{\prime }& =\underset{h\to 0}{\lim }\frac{f(x+h)g(x+h)-f(x)g(x)}{h}\\ & =\underset{h\to 0}{\lim }\frac{[f(x+h)-f(x)]g(x+h)+f(x)[g(x+h)-g(x)]}{h}\\ & =f^{\prime }(x)g(x)+f(x)g^{\prime }(x).\end{array}$$ $$[f(x)g(x){]}^{\prime }=f^{\prime }(x)g(x)+f(x)g^{\prime }(x).$$

Quotient Rule

The quotient rule gives the derivative of the ratio of two differentiable functions. Starting from the limit definition and adding and subtracting $f(x)g(x)$ in the numerator:

$$\begin{array}{rl}{\left(\frac{f(x)}{g(x)}\right)}^{\prime }& =\underset{h\to 0}{\lim }\frac{f(x+h)g(x)-f(x)g(x+h)}{hg(x+h)g(x)}\\ & =\frac{1}{g(x{)}^2}\left[f^{\prime }(x)g(x)-f(x)g^{\prime }(x)\right].\end{array}$$ $${\left(\frac{f(x)}{g(x)}\right)}^{\prime }=\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{g(x{)}^2}.$$

Applications

• Computing derivatives of polynomials, rational functions, and algebraic expressions.
• Building blocks for differentiating more complex functions together with the chain rule.
• The quotient rule underlies the derivatives of $\tan x$, $\cot x$, $\sec x$, $\csc x$.

Trade-offs

• The power rule in its basic form requires integer (or later rational/real) exponents; for negative or fractional powers, the general form via $\exp (p\ln x)$ requires $x>0$.
• The quotient rule requires $g(x)\ne 0$ at the point of differentiation.
• The product and quotient rules assume both $f$ and $g$ are differentiable at the point.

Links

• Definition of the Derivative
• Chain Rule
• Common Derivatives