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Differentiation techniques — Formula Sheet

4 formulas from Calculus I, each with what it means, when to use it and the trap it sets.

Product rule

(f g)^{'} = f^{'} g + f g^{'}

What it says: Differentiate the first and keep the second, then keep the first and differentiate the second.
When to reach for it: Two genuinely different functions multiplied together.
Watch out: The derivative of a product is not the product of the derivatives. (fg)' ≠ f'g'.

\frac{d}{d x} (x^{2} sin x) = 2 x sin x + x^{2} cos x

Two terms out, always — one for each factor taking its turn.

Quotient rule

(\frac{f}{g})^{'} = \frac{f ^{'} g - f g ^{'}}{g ^{2}}

What it says: Bottom times derivative of the top, minus top times derivative of the bottom, all over the bottom squared.
When to reach for it: A fraction whose denominator actually contains the variable.
Watch out: The order in the numerator matters — the minus makes it non-commutative. Swap the terms and you get the negative of the answer.

\frac{d}{d x} \frac{x}{x + 1} = \frac{( x + 1 ) - x}{( x + 1 ) ^{2}} = \frac{1}{( x + 1 ) ^{2}}

Often quicker to rewrite as a product with a negative power instead.

Chain rule

\frac{d}{d x} f (g (x)) = f^{'} (g (x)) g^{'} (x)

What it says: Differentiate the outside leaving the inside alone, then multiply by the derivative of the inside.
When to reach for it: Anything nested — a bracket raised to a power, a trig function of an expression, e to the something.
Watch out: The missing inner derivative is the most common error in the whole course. If you named an inside function, its derivative must appear in your answer.

\frac{d}{d x} (3 x^{2} + 1)^{5} = 5 (3 x^{2} + 1)^{4} \cdot 6 x

The 6x is the inner derivative; without it the answer is silently wrong.

Implicit differentiation

\frac{d}{d x} [y^{n}] = n y^{n - 1} \frac{d y}{d x}

What it says: Differentiate both sides in x, and every y term hands you a dy/dx by the chain rule.
When to reach for it: Equations you cannot or would rather not rearrange into y = something.
Watch out: Forgetting the dy/dx on a y term is exactly the chain-rule error wearing a different hat.

x^{2} + y^{2} = 25 \Rightarrow \frac{d y}{d x} = - \frac{x}{y}

Collect every dy/dx on one side, then divide.

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