0.5: Differentiating Elementary Functions
0.5.1 The Power Rule
The derivative of a power of \(x\) is given by
\[\frac{d}{dx}x^p=px^{p-1}.\nonumber \]
0.5.2 Trigonometric Functions
The derivatives of \(\sin x\) and \(\cos x\) are
\[(\sin x)'=\cos x,\quad (\cos x)'=-\sin x.\nonumber \]
We thus say that “the derivative of sine is cosine,” and “the derivative of cosine is minus sine.” Notice that the second derivatives satisfy
\[(\sin x)''=-\sin x,\quad (\cos x)''=-\cos x.\nonumber \]
0.5.3 Exponential and Natural Logarithm Functions
The derivative of \(e^x\) and \(\ln x\) are
\[(e^x)'=e^x,\quad (\ln x)'=\frac{1}{x}.\nonumber \]