Derivative of Power of Function

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Theorem

Let $\map u x$ be a differentiable real function of $x$.

Let $n$ be a real number such that $n \ne -1$.

Then:

$\map {\dfrac \d {\d x} } {\map u x^n} = n \map u x^{n - 1} \map {\dfrac \d {\d x} } {\map u x}$


Proof 1

\(\ds \map {\frac \d {\d x} } {\map u x^n}\) \(=\) \(\ds \map {\frac \d {\d u} } {\map u x^n} \map {\frac \d {\d x} } {\map u x}\) Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds n \map u x^{n - 1} \map {\frac {\d u} {\d x} } {\map u x}\) Derivative of Hyperbolic Sine

$\blacksquare$


Proof 2

\(\ds \map {\dfrac \d {\d x} } {\map u x^n}\) \(=\) \(\ds \lim_{h \mathop \to 0} \frac {\paren {\map u {x + h} }^n - \paren {\map u x}^n} h\)
\(\ds \) \(=\) \(\ds \paren {\map u x}^n \lim_{h \mathop \to 0} \frac {\paren {\frac {\map u {x + h} } {\map u x} }^n - 1} h\) Power of Product
\(\ds \) \(=\) \(\ds \paren {\map u x}^n \lim_{h \mathop \to 0} \frac {\exp \paren {n \ln \frac {\map u {x + h} } {\map u x} } - 1} h\) Definition of Power to Real Number
\(\ds \) \(=\) \(\ds \paren {\map u x}^n \lim_{h \mathop \to 0} \paren {\frac {\map \exp {n \ln \frac {\map u {x + h} } {\map u x} } - 1} {n \ln \frac {\map u {x + h} } {\map u x} } } \paren {\frac {n \ln \frac {\map u {x + h} } {\map u x} } h}\)
\(\ds \) \(=\) \(\ds \paren {\map u x}^n \lim_{h \mathop \to 0} \frac {n \ln \frac {\map u {x + h} } {\map u x} } h\) Derivative of Exponential at Zero
\(\ds \) \(=\) \(\ds n \paren {\map u x}^n \lim_{h \mathop \to 0} \frac {\ln \frac {\map u {x + h} } {\map u x} } h\)
\(\ds \) \(=\) \(\ds n \paren {\map u x}^n \lim_{h \mathop \to 0} \frac {\map \ln {1 + \frac {\map u {x + h} - \map u x} {\map u x} } } h\)
\(\ds \) \(=\) \(\ds n \paren {\map u x}^n \lim_{h \mathop \to 0} \paren {\frac {\map \ln {1 + \frac {\map u {x + h} - \map u x} {\map u x} } } {\frac {\map u {x + h} - \map u x} {\map u x} } } \paren {\frac {\frac {\map u {x + h} - \map u x} {\map u x} } h }\)
\(\ds \) \(=\) \(\ds n \paren {\map u x}^n \lim_{h \mathop \to 0} \frac {\paren {\frac {\map u {x + h} - \map u x} {\map u x} } } h\) Derivative of Logarithm at One
\(\ds \) \(=\) \(\ds n \paren {\map u x}^n \lim_{h \mathop \to 0} \frac 1 {\map u x} \frac {\map u {x + h} - \map u x} h\)
\(\ds \) \(=\) \(\ds n \paren {\map u x}^{n - 1} \lim_{h \mathop \to 0} \frac {\map u {x + h} - \map u x} h\) Product of Powers
\(\ds \) \(=\) \(\ds n \paren {\map u x}^{n - 1} \map {\dfrac \d {\d x} } {\map u x}\)

$\blacksquare$


Also presented as

This can be (and usually is) presented more simply as:

$\map {\dfrac \d {\d x} } {u^n} = n u^{n - 1} \dfrac {\d u} {\d x}$


Sources

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