Derivatives Explained: Rules, Interpretation, and Applications

Definition from Limits

The derivative f'(x) is defined as a limit: the limit of the difference quotient (average rate of change) as the interval shrinks to zero.

Geometrically, this is the slope of the tangent line at x. Physically, it's the instantaneous rate of change — velocity if f(x) is position, or marginal cost if f(x) is total cost. Every differentiation rule in the next sections is derived from this single definition.

The Power Rule

The most-used differentiation rule applies to any power of x:

Bring the exponent down as a coefficient, then subtract 1 from the exponent. Works for any real exponent — positive, negative, or fractional.

Examples:

Fractional exponents work the same way — rewrite radicals as fractional powers first:

The Product Rule

When differentiating a product, you can't just multiply the derivatives. The correct rule:

Mnemonic: "derivative of the first times the second, plus the first times derivative of the second."

Example:

The Quotient Rule

For quotients, the rule has a subtraction and a squared denominator:

Mnemonic: "low d-high minus high d-low, over the square of what's below." Order matters here — unlike the product rule, the quotient rule subtracts.

Example:

The Chain Rule

For composite functions, the chain rule says: differentiate the outer function, keep the inside unchanged, then multiply by the derivative of the inside.

Simple example:

The outer derivative is cos; the inner derivative is 3. Multiply them.

Combined with power rule:

The chain rule is the most commonly-forgotten rule. Whenever you differentiate something inside parentheses, brackets, a function name, or a root — you need the chain rule.

Tangent Lines and Geometric Interpretation

The derivative at a point equals the slope of the tangent line there. The tangent line itself has equation:

Example: Find the tangent line for f(x) = x^2 \text{ at } a = 3. Since f'(x) = 2x, the slope at x = 3 is 6. The point is (3, 9):

This interpretation turns derivatives into a powerful approximation tool — locally, any smooth function looks like its tangent line (the linear approximation).

Optimization: Finding Maxima and Minima

A function's derivative is zero at local maxima and minima (where the tangent is horizontal). To find them:

1. Compute f'(x).
2. Solve f'(x) = 0 for critical points.
3. Classify each critical point as max, min, or saddle using the first or second derivative test.
4. Check endpoints if the domain is bounded.

Example: Find critical points of f(x) = x^3 - 3x^2.

Critical points at x = 0 and x = 2. Testing shows x = 0 is a local max and x = 2 is a local min — this is how derivatives solve optimization problems across physics, economics, and engineering.

Common Mistakes

Practice Problems

Find the derivative of each function. Identify which rules you need.

1. f(x) = 4x^3 - 2x + 7
2. f(x) = x^2 \cos x
3. f(x) = \dfrac{x+1}{x-1}
4. f(x) = (x^2+3x)^4
5. f(x) = \sin(x^2)
6. f(x) = \ln(x^2+1)
7. f(x) = \sqrt{3x+5}

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