Derivative of Power of Function/Proof 2
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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
| \(\ds \map {\dfrac \d {\d x} } {\map u x^n}\) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {\paren {\map u {x + h} }^n - \paren {\map u x}^n} h\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\map u x}^n \lim_{h \mathop \to 0} \frac {\paren {\frac {\map u {x + h} } {\map u x} }^n - 1} h\) | 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 | |||||||||||
| \(\ds \) | \(=\) | \(\ds n \paren {\map u x}^{n - 1} \map {\dfrac \d {\d x} } {\map u x}\) |
$\blacksquare$