Differentiation

What You Will Learn in This Chapter

Differentiation is one of the two main operations of calculus. The 15 sections below take you from the question "when does a derivative fail to exist?" all the way to differentials and their applications.

Section	Topic	Key Ideas
3.1	When a Function Is Not Differentiable	Corners, cusps, vertical tangents, oscillation
3.2	Power Rule	$\frac{d}{dx}{x}^r=r{x}^{r-1}$ for all real $r$
3.3	The Algebra of Derivatives	Sum, Constant Multiple, Product, Quotient Rules
3.4	The Chain Rule	Differentiating composite functions
3.5	Derivatives of Trigonometric Functions	$\sin x$, $\cos x$, $\tan x$, and more
3.6	Higher Derivatives	Second, third, and $n$-th derivatives
3.7	More about the Leibniz Notation	Second-increment quotient interpretation
3.8	Implicit Differentiation	Curves defined by $F(x,y)=0$
3.9	Derivatives of Inverse Functions	The Derivative Rule for Inverses
3.10	Derivatives of Inverse Trig Functions	$\arcsin$, $\arccos$, $\arctan$
3.11	Derivatives of Logarithmic Functions	$\ln x$, ${\log }_{a}x$, logarithmic differentiation
3.12	Derivatives of Exponential Functions	$e^x$, $a^x$, $u^v$
3.13	Hyperbolic Functions and Their Derivatives	$\sinh$, $\cosh$, $\tanh$, and inverses
3.14	Linear Approximations	Tangent-line approximation $f(x)\approx L(x)$
3.15	Differentials	dy = f'(x)\,dx and propagation of error

Real-World Applications

• Physics: Velocity and acceleration are first and second derivatives of position.
• Engineering: Linear approximations simplify complex equations (pendulum problem).
• Economics: Marginal cost and marginal revenue are derivatives of cost and revenue functions.
• Biology: Growth rate models use exponential and logarithmic derivatives.
• Error analysis: Differentials estimate how measurement errors propagate through formulas.

Frequently Asked Questions

What is a derivative, intuitively?

The derivative f'(x) is the instantaneous rate of change of $f$ at $x$. Geometrically, it is the slope of the tangent line to the graph of $f$ at the point $(x,f(x))$.

What is the difference between differentiability and continuity?

Every differentiable function is continuous, but not every continuous function is differentiable. For example, $f(x)=|x|$ is continuous at $x=0$ but has a corner there, so it is not differentiable at $x=0$.

Why are there so many differentiation rules?

Each rule handles a different way functions can be combined: addition, multiplication, division, composition, and inversion. Together these rules let you differentiate any function built from the standard elementary functions.

What is a differential and how does it differ from a derivative?

The derivative $dy/dx$ is a single number (the rate of change). A differential dy = f'(x)\,dx treats $dx$ as an independent variable; their ratio equals the derivative when $dx\ne 0$. Differentials are especially useful for estimating errors and for substitution in integrals.

When should I use implicit differentiation?

Use implicit differentiation when a relation $F(x,y)=0$ cannot be solved explicitly for $y$ (or when doing so would be very messy). You differentiate both sides with respect to $x$, treating $y$ as a function of $x$ and applying the Chain Rule wherever $y$ appears.