3 The Derivative and Differentiation Rules

From Average Change to the Derivative

The central idea is instantaneous change. For a function $f$, the average rate of change from $x=a$ to $x=a+h$ is

This is the slope of a secant line through two points on the graph. Letting $h$ approach zero moves the second point toward the first and produces the slope of the tangent line:

Replacing $a$ with the variable $x$ gives the derivative function:

A limit-based example

For $f(x)=x^2$, simplify the difference quotient before taking the limit:

Direct substitution into the unsimplified quotient would produce the indeterminate form $\frac{0}{0}$, so algebraic simplification comes first.

Takeaway: The derivative is the limit of average rates of change as the input interval shrinks to zero.

Meaning and Notation

The derivative has both geometric and applied meanings. At the point $(a,f(a))$, $f'(a)$ is the slope of the tangent line. A positive derivative indicates that the function is increasing locally, a negative derivative indicates local decrease, and a zero derivative gives a horizontal tangent.

If $y=f(x)$, common notations are $f'(x)$, $y'$, $\frac{dy}{dx}$, and $\frac{d}{dx}[f(x)]$. The notation $f'(a)$ means the value of the derivative at one input, whereas $f'(x)$ denotes the whole derivative function.

Derivatives also describe rates with units. If $s(t)$ is position as a function of time, then

is instantaneous velocity. The units of a derivative are output units divided by input units.

Differentiability is a stronger condition than continuity: differentiability at a point implies continuity there, but continuity does not always imply differentiability. The function $f(x)=|x|$ is continuous at $x=0$, yet it has different one-sided slopes and is not differentiable at that point.

Takeaway: Interpret a derivative as a slope, an instantaneous rate, and a quantity whose units reflect the ratio of output change to input change.

Core Differentiation Rules

The basic rules turn limit definitions into efficient calculations.

Core algebraic rules

For a constant $C$,

For a constant multiple,

Sums and differences can be differentiated term by term:

The Power rule gives

For instance,

Products and quotients

Use the Product rule for a product:

Use the quotient rule for a quotient with $g(x)\ne 0$:

For example,

Takeaway: First identify the outer algebraic structure—constant, sum, product, or quotient—then apply the matching rule carefully.

Common Derivative Formulas

Several derivative formulas serve as a reference set:

• $\frac{d}{dx}[e^x]=e^x$

• $\frac{d}{dx}[a^x]=a^x\ln a$, where $a>0$ and $a\ne 1$

• $\frac{d}{dx}[\ln x]=\frac{1}{x}$

• $\frac{d}{dx}[\sin x]=\cos x$

• $\frac{d}{dx}[\cos x]=-\sin x$

• $\frac{d}{dx}[\tan x]=\sec^2 x$

• $\frac{d}{dx}[\cot x]=-[\csc x]^2$

• $\frac{d}{dx}[\sec x]=\sec x\tan x$

• $\frac{d}{dx}[\csc x]=-[\csc x][\cot x]$

For logarithms with another base, use

which gives

Two useful inverse-trigonometric formulas are

Takeaway: Recognizing a function from a standard derivative table reduces computation to selecting the correct formula and checking its domain.

Compositions and the Chain Rule

A composition has one function inside another, such as $\sin(x^2)$, $(3x-1)^5$, or $e^{4x+2}$. The Chain rule handles these nested structures. If $h(x)=f(g(x))$, then

A reliable process is:

1. Identify the outer function and the inner function.

2. Differentiate the outer function while keeping the inner expression unchanged.

3. Multiply by the derivative of the inner function.

For example,

For a trigonometric composition,

In Leibniz notation, if $y=f(u)$ and $u=g(x)$, then

Takeaway: Nested functions require both the derivative of the outside layer and the derivative of the inside layer.

Implicit Relationships and Slopes

Use Implicit differentiation when an equation relates $x$ and $y$ without isolating $y$. Differentiate both sides with respect to $x$, treat $y$ as a function of $x$, collect terms containing $\frac{dy}{dx}$, and solve.

Circle example

For

differentiate both sides:

Solving gives

At $(3,4)$, the slope is

The tangent line is therefore

Product involving the dependent variable

For

differentiate using the Product rule and the Chain rule:

Group the derivative terms and solve:

Takeaway: Every occurrence of a dependent variable such as $y$ contributes a factor of $\frac{dy}{dx}$ when differentiated with respect to $x$.