AP Calculus - Derivative Definition & Rules

The derivative of a function f at a point x=a is defined as: $$f'(a) = \frac{f(a + \text{h}) - f(a)}{\text{h}} \text{ as } \text{h} \to 0$$ .

$$\frac{d}{dx}[\text{arccsc}(x)] = -\frac{1}{x\text{sqrt}(x^2-1)}$$ .

$$f'(a) = \frac{f(x) - f(a)}{x - a} \text{ as } x \to a$$ .

1. Log, 2. Simplify, 3. Differentiate.

$$\frac{d}{dx}[\text{sec}(x)] = \text{sec}(x)\tan(x)$$ .

$$\frac{d}{dx}[b^x] = \text{ln}(b) \times b^x$$ .

If $$f(x) = x^n$$ , then $$f'(x) = n \times x^{n-1}$$ .

If $$f(x) = u(x) \times v(x)$$ , then $$f'(x) = u'(x)v(x) + u(x)v'(x)$$ .

If $$f(x) = \frac{u(x)}{v(x)}$$ , then $$f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$$ .

If $$f(x) = g(h(x))$$ , then $$f'(x) = g'(h(x)) \times h'(x)$$ .