Differentiation

What a derivative measures

Differentiation is the process of finding a derivative. If a function is written as $y=f(x)$, common notations for its derivative are

The derivative has two closely related interpretations:

• Geometrically, it is the slope of the tangent line to a curve at a point.

• In an application, it is the instantaneous rate of change of one quantity with respect to another.

For example, if position is a function of time, its derivative is instantaneous velocity. The derivative is therefore a local measurement: it describes what is happening near a particular input rather than over a broad interval.

Takeaway: Differentiation converts a function into a new function that records local slope or instantaneous change.

The derivative from a limit

A secant line passes through two points on the graph of $y=f(x)$. If the inputs differ by $h$, its slope is the difference quotient

As the second point moves toward the first, $h\to 0$, the secant slope approaches the slope of the tangent line. This gives the limit definition of the derivative:

An equivalent form at the input $a$ is

The derivative exists at $x=a$ only when this limit exists as a finite number. At an endpoint, a one-sided limit may be used.

Example: Finding the derivative of $f(x)=x^2$

Thus, the slope at $x=a$ is $2a$, and the tangent line at $(a,a^2)$ is

Takeaway: The derivative is obtained by simplifying the average-rate expression and then taking its limit as the input change approaches zero.

Differentiability and continuity

Differentiability and continuity are related but not equivalent. If a function is differentiable at $a$, then it is continuous at $a$. However, continuity alone does not guarantee differentiability.

A function can fail to be differentiable at a:

• corner,

• cusp,

• vertical tangent, or

• discontinuity.

For example, $f(x)=|x|$ is continuous at $0$, but it has no derivative there. The slope from the left is $-1$, while the slope from the right is $1$; because the one-sided slopes do not agree, the derivative does not exist.

Takeaway: Differentiability implies continuity, but a continuous function may still have a sharp feature where no single tangent slope exists.

Core differentiation rules

Basic derivative rules make it possible to differentiate complicated expressions efficiently.

• Power rule:

  $$\frac{d}{dx}[x^n]=nx^{n-1}.$$

• Constant rule:

  $$\frac{d}{dx}[c]=0.$$

• Constant-multiple rule:

  $$\frac{d}{dx}[cf(x)]=cf'(x).$$

• Sum and difference rules:

  $$\frac{d}{dx}[f(x)\pm g(x)]=f'(x)\pm g'(x).$$

• Product rule:

  $$\frac{d}{dx}[uv]=u'v+uv'.$$

• Quotient rule:

  $$\frac{d}{dx}\left[\frac{u}{v}\right]=\frac{vu'-uv'}{v^2}.$$

For example,

For an expression that is a composition, use the chain rule:

The generalized power rule is a common chain-rule application:

Takeaway: First identify the structure of the expression, then select the rule that matches that structure.

Algebraic functions

Rewrite radicals and reciprocals as powers when this makes the rules easier to apply:

For example,

Algebraic functions may also require several rules at once. Consider

Using the product rule and the chain rule gives

A factored derivative is often useful because it can make zeros, factors, and sign changes easier to see. Always keep domain restrictions in mind when radicals occur in denominators or when negative powers are used.

Takeaway: Converting algebraic expressions to powers often exposes the product, quotient, and composition structure needed for differentiation.

Trigonometric, exponential, and logarithmic derivatives

When differentiating trigonometric, exponential, or logarithmic functions, combine the relevant basic formula with the chain rule when the input is not simply $x$. Trigonometric derivative formulas assume angles are measured in radians.

Trigonometric functions

For example,

Exponential functions

and for $a>0$, $a\ne 1$,

Thus,

Logarithmic functions

where $u(x)\ne 0$. In particular,

For a logarithm with base $a$,

Takeaway: Memorize the basic derivative families, then multiply by the derivative of the inner expression whenever the input is composite.

Inverse functions

The derivative of an inverse function can be found from the derivative of the original function. If $f$ is one-to-one and differentiable, then

provided the denominator is nonzero.

Important inverse trigonometric formulas include

and

With a composite argument,

For example,

The formula for $\arcsin x$ can be derived by setting $y=\arcsin x$, so that $\sin y=x$. Differentiating implicitly gives $\cos y\,dy/dx=1$, and the principal range of $\arcsin$ supplies $\cos y=\sqrt{1-x^2}$.

Takeaway: Inverse-function derivatives are reciprocals of the corresponding original-function derivatives, evaluated at the matching inverse input.

Finding slopes implicitly

Use implicit differentiation when an equation relates $x$ and $y$ but does not conveniently give $y$ as an explicit function of $x$. Differentiate both sides with respect to $x$, treat $y$ as a function of $x$, apply the chain rule to terms involving $y$, and solve for $dy/dx$.

For example, given the circle

differentiation gives

Therefore,

At $(3,4)$, the slope is $-3/4$, so the tangent line is

For a more complicated relation,

differentiation gives

Collecting the derivative terms yields

Takeaway: Every occurrence of $y$ must be differentiated as a function of $x$; this is why factors of $dy/dx$ appear.

Higher derivatives and a differentiation strategy

A second derivative is the derivative of the first derivative:

More generally, repeated differentiation produces higher derivatives. For

we obtain

The first derivative describes instantaneous rate of change. The second derivative describes how that rate changes. In motion problems, if position is differentiated once to obtain velocity, differentiating again gives acceleration.

For a complicated function, use this workflow:

1. Identify the outermost operation: sum, product, quotient, or composition.

2. Rewrite radicals and reciprocals as powers when helpful.

3. Apply the appropriate structural rule.

4. Differentiate the basic pieces.

5. Simplify only when simplification improves clarity.

6. Check restrictions from logarithms, denominators, radicals, and inverse trigonometric functions.

For example, with

use the product rule and the chain rule:

The real-valued function requires $\sin x>0$, and the derivative is defined only where the entire expression is defined.

Takeaway: A reliable differentiation strategy is structural: identify the outer operation, apply the matching rule, differentiate inward, and verify the domain.

• logarithmic differentiation