Worked Derivative Solutions

Every page in this library walks one derivative from start to finish: the rule that applies, each step named and explained, the mistakes students actually make, and a few practice problems to lock it in. Each solution links straight into the calculator with the function pre-loaded, so you can experiment with variations of your own.

If you're just starting out, begin with the power rule pages — they're the foundation everything else builds on. The chain rule section is where most students hit their first real wall; the sin(2x) walkthrough is the gentlest entry point.

Power rule

One rule, four disguises. Whole-number powers are the warm-up; roots and reciprocals are the same rule wearing fractional and negative exponents.

x²= 2x x³= 3x2 √x= 1/(2√x) 1/x= −1/x2

Trigonometric functions

The sine and cosine results are memorized; tangent and secant are derived from them with the quotient rule — each page shows the derivation, not just the answer.

sin(x)= cos(x) cos(x)= −sin(x) tan(x)= sec2(x) sec(x)= sec(x) tan(x)

Chain rule

Composite functions: differentiate the outside, keep the inside, multiply by the inner derivative. These five are the most common exam variants — including one where the chain rule's factor cancels away entirely.

sin(2x)= 2 cos(2x) cos(2x)= −2 sin(2x) tan(2x)= 2 sec2(2x) e^(2x)= 2e2x ln(2x)= 1/x

Exponentials & logarithms

Why e is special, what every other base pays for not being e, and why the natural log's derivative contains no logarithm at all.

e^x= ex 2^x= 2x ln 2 ln(x)= 1/x log(x)= 1/(x ln 10)

Inverse trigonometric

Derived by implicit differentiation, these answers contain no trig — just rational functions and square roots that explain each curve's shape.

arctan(x)= 1/(1 + x2) arcsin(x)= 1/√(1 − x2)

Product rule & advanced

A true product that no shortcut can touch, and the famous x^x — the function that breaks both the power rule and the exponential rule and forces logarithmic differentiation.

x·sin(x)= sin(x) + x cos(x) x^x= xx(ln x + 1)