3.2 | Basic Differentiation Rules and Rates of Change

The Constant Rule: The derivative of a constant function is $$0$$. That is, if $$c$$ is a real number, then
$\frac{d}{dx}[c] = 0.$
The Power Rule: If $$n$$ is a rational number, then the function $$f(x) = x^n$$ is differentiable and
$$\frac{d}{dx}[x^n] = n x^{n-1}.$$
For $$f$$ to be differentiable at $$x = 0$$, $$n$$ must be a number such that $$x^{n-1}$$ is defined on the interval containing 0.
The Constant Multiple Rule: If $$f$$ is a differentiable function and $$c$$ is a real number, then $$c f$$ is also differentiable and
$\frac{d}{dx}[cf(x)] = c f'(x).$
The Sum and Difference Rules: The sum (or difference) of two differentiable functions $$f$$ and $$g$$ is itself differentiable. Moreover, the derivative $$f+g$$ (or $$f-g$$ is the sum (or difference) of the derivatives of $$f$$ and $$g$$.
$$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$$
Derivatives of Sine and Cosine Functions
$$\frac{d}{dx}[\sin x] = \cos x$$
$$\frac{d}{dx}[\cos x] = -\sin x$$
Derivative of the Natural Exponential Function
$$\frac{d}{dx}[e^x] = e^x$$