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
$$
\frac{d}{dx}[e^x] = e^x
$$

E 3.2 Exercises