Derivative meaning, notation, and the core differentiation rules

Key idea

The derivative measures how fast a function changes. For a graph \(y=f(x)\), the derivative \(f'(x)\) is the slope of the tangent line. Common notations mean the same thing: \[ f'(x),\quad \frac{dy}{dx},\quad \frac{d}{dx}[f(x)]. \] In calculus, the derivative can be defined using a limit (difference quotient): \[ f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}, \] but most practice problems are solved using the differentiation rules below.

Rules you’ll use constantly

• Constant rule: \(\dfrac{d}{dx}[c]=0\)
• Power rule: \(\dfrac{d}{dx}[x^n]=n x^{n-1}\) (works for integers, fractions, and negatives)
• Sum/difference: \((f\pm g)'=f'\pm g'\)
• Constant multiple: \((cf)'=c f'\)

Worked example

Try it

Summary

• Differentiate term-by-term using linearity: constants, sums, and constant multiples.
• The power rule works for negative and fractional exponents (where the function is defined).

Chain rule for composite functions (inside-out differentiation)

Key idea

A composite function has an “inside” function and an “outside” function. If \(y=f(g(x))\), then the chain rule says: \[ \frac{dy}{dx}=f'(g(x))\cdot g'(x). \] A quick workflow: identify the inside \(u=g(x)\), differentiate the outside with respect to \(u\), then multiply by \(\dfrac{du}{dx}\).

Worked example

Try it

Summary

• Composite function? Use the chain rule: outside derivative \(\times\) inside derivative.
• Rewrite radicals and fractions as powers when it simplifies the chain rule.

Trigonometric derivatives and common compositions

Core trig derivatives (memorize these)

• \((\sin x)'=\cos x\)
• \((\cos x)'=-\sin x\)
• \((\tan x)'=\sec^2 x\)
• \((\csc x)'=-\csc x\cot x\)

Worked example

Try it

Summary

• Memorize trig derivatives, then apply chain rule to anything like \(\sin(2x)\) or \((\tan x)^2\).
• Be careful with signs: \((\cos x)'=-\sin x\) and \((\csc x)'=-\csc x\cot x\).

Product rule and quotient rule (plus smart rewriting)

Key rules

• Product rule: \((uv)'=u'v+uv'\)
• Quotient rule: \(\left(\dfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^2}\), v≠ 0

Worked example

Try it

Summary

• Use the product rule for products; use the quotient rule for quotients when rewriting isn’t simpler.
• Algebra first can save time: \(\dfrac{x^2+1}{x}=x+\dfrac{1}{x}\).

Exponential and logarithmic derivatives (plus chain rule composites)

Core rules (memorize these)

• \((e^x)'=e^x\)
• \((ae^x)'=ae^x\) for constant \(a\)
• \((\ln x)'=\dfrac{1}{x}\) (for \(x>0\))

Worked example

Try it

Summary

• Exponentials: \((e^u)'=e^u\cdot u'\).
• Logs: \((\ln u)'=\dfrac{u'}{u}\) (where \(u>0\)).
• Many “hard” derivatives are just chain rule with these base formulas.

Fast strategy: choose the right rule and avoid common mistakes

Key idea

Most derivative problems become easy if you first identify the structure: sum/difference, constant multiple, power, product, quotient, or composite. When possible, rewrite to a simpler form: \[ \frac{1}{x}=x^{-1},\qquad \sqrt{x+1}=(x+1)^{1/2},\qquad \frac{x^2+1}{x}=x+\frac{1}{x}. \] Then apply the smallest set of rules needed.

Worked example (smart rewrite)

Try it

Summary

• Always identify structure first: sum, product, quotient, or composite.
• Rewrite when helpful to reduce quotient rule into a simpler power rule problem.
• Chain rule is the most common reason answers are missing a factor (the inside derivative).

Why derivatives matter (and a final practice check)

Where derivatives show up

• Tangent lines: slope at a point and linear approximation.
• Rates of change: velocity, acceleration, growth/decay, marginal cost/revenue.
• Optimization: maximizing profit, minimizing time, designing efficient shapes.
• Curve analysis: increasing/decreasing, concavity, and sketching graphs.

Worked example: multiple rules in one problem

Try it

Final recap

• Core rules: constants \(\to 0\), powers \(\to nx^{n-1}\), linearity for sums and constant multiples.
• Chain rule: the most common rule in composite functions like \((3x-2)^4\), \(\cos(2x-1)\), \(e^{x^2}\), \(\ln(\sin x)\).
• Product/quotient: use \((uv)'=u'v+uv'\) and \(\left(\dfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^2}\) (or simplify first).
• Trig/exp/log: memorize the base derivatives, then apply chain rule when the input is not just \(x\).