1.5: Differentiating Elementary Functions
The Power Rule
The derivative of a power of \(x\) is given by \[\frac{d}{dx}x^p=px^{p-1}.\nonumber\]
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 \[(\sin x)''=-\sin x,\quad (\cos x)''=-\cos x.\nonumber\]
Exponential and Natural Logarithm Functions
The derivative of \(e^x\) and \(\ln x\) are \[(e^x)'=e^x,\quad (\ln x)'=\frac{1}{x}.\nonumber\]