Gradient function = $\frac{dy}{dx}$ (or $\frac{d}{dx}y$, or $f'(x)$) — no tangents needed.

Reflex: Multiply by power, power reduce by 1.

Convert back after: $-2x^{-3} = -\frac{2}{x^3}$, $\frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$.

Reflex: Indices in, differentiate, indices out.

Differentiate the outside function, times differentiate the inside function.

Reflex: Outside first, then × inside derivative — never forget the inside.

A negative or fractional outer power works exactly like an integer one — multiply by it, reduce by 1, then × inside derivative.

Reflex: Reciprocal → negative power; root → fractional power; then outside × inside.

Use when two genuinely different functions of $x$ multiply and can't be expanded away. Reflex: Diff first × keep second + diff second × keep first.

Common bracket, different powers → pull out the lowest (most negative) one. Never expand both terms separately.

Reflex: Common bracket → pull out the lowest (most negative) power, then tidy what's left.

Diff top keep bottom, minus diff bottom keep top, all over bottom². Minus sign means order matters. Reflex: diff top·bottom − diff bottom·top, over bottom².

Divide out, factorize-cancel, or expand before reaching for product/quotient rule — a 10-second algebra step often kills a messy computation.

Reflex: Always simplify first if you can; remember the chain rule.

For $f(x)=x^2$: $\frac{(x+\delta x)^2 - x^2}{\delta x} = 2x + \delta x \to 2x$ — multiply by power, reduce by 1. Reflex: $\frac{dy}{dx}$ is the limit of chord gradients as the chord shrinks to a point.

Linear inside multiplies by $a$. Powers: $\sin^n,\tan^n$ stay positive; only $\cos^n$ flips sign. Reflex: Only cos brings a minus; tan brings sec².

Each layer — power, trig, inside — gives one multiplier.

Reflex: One multiplier per layer — count your layers before you start.

Exponential survives, inside derivative multiplies: $3e^{4x^2+3x}(8x+3)$. Power wraps e^f? Do the power first, then chain inside. Reflex: $e^f$ never disappears — copy it, then × inside derivative.

Simplify first: powers drop down, quotients split (minus!), roots become half-powers — then differentiate the easy pieces. Reflex: Log laws first; quotient-split carries a minus on the denominator's log.