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What a derivative is — Formula Sheet

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

Limit definition of the derivative

f^{'} (a) = h \to 0 lim \frac{f ( a + h ) - f ( a )}{h}

What it says: The slope between two points on the curve, as the second point slides into the first.
When to reach for it: When a question says "from first principles", and whenever you need to know why a rule works.
Watch out: Simplify and cancel the h before letting h go to zero — substituting first only ever gives 0/0.

f (x) = x^{2} \Rightarrow f^{'} (a) = h \to 0 lim (2 a + h) = 2 a

Every surviving term contained h, so it divided out before the limit.

Tangent line at a point

y - f (a) = f^{'} (a) (x - a)

What it says: The straight line that touches the curve at a and matches its slope there.
When to reach for it: Any "find the equation of the tangent" question, and linear approximation.
Watch out: f'(a) is the slope, f(a) is the height. Swapping them is the standard slip.

y - 4 = 4 (x - 2) \Rightarrow y = 4 x - 4

Tangent to y = x² at x = 2.

Power rule

\frac{d}{d x} x^{n} = n x^{n - 1}

What it says: Bring the power down in front, then knock one off the power.
When to reach for it: Every polynomial, and every root or reciprocal once you rewrite it as a power.
Watch out: It applies to a variable base with a constant power. 2^x is not covered by it.

\frac{d}{d x} x = \frac{d}{d x} x^{1/2} = \frac{1}{2} x^{- 1/2}

Rewriting the root as a power is what lets the rule reach it.

Trigonometric derivatives

(sin x)^{'} = cos x, (cos x)^{'} = - sin x, (tan x)^{'} = sec^{2} x

What it says: Sine and cosine cycle into each other; only cosine picks up the minus sign.
When to reach for it: Any trig term, and as the outer function in most chain-rule questions.
Watch out: These hold for radians only. In degrees every one of them gains a factor of π/180.

\frac{d}{d x} sin (3 x) = 3 cos (3 x)

The 3 is the inner derivative — the rule alone would have missed it.

Exponential and log derivatives

(e^{x})^{'} = e^{x}, (ln x)^{'} = \frac{1}{x}, (a^{x})^{'} = a^{x} ln a

What it says: e^x is its own derivative; every other base pays a factor of ln a for the privilege.
When to reach for it: Growth and decay models, and anything with e or ln in it.
Watch out: 1/x is the derivative of ln x, and it is only defined for x > 0.

\frac{d}{d x} ln (5 x) = \frac{5}{5 x} = \frac{1}{x}

The chain rule contributes the 5, which then cancels.

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