微积分-常用导数(二):最常用到的技巧

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指数函数求导:

Q:\frac{\mathrm{d} }{\mathrm{d} x} a^x=?

A:

\begin{aligned} a=e^{\ln_{}{a} }\\ a^x=(e^{\ln_{}{a} })^x=e^{x\ln_{}{a} }\\ \frac{\mathrm{d} }{\mathrm{d} x} a^x=\frac{\mathrm{d} }{\mathrm{d} x}e^{x\ln_{}{a} } =\ln_{}{a}·e^{x\ln_{}{a}}=a^x\ln_{}{a} \end{aligned}

或者(链式法则):

Q:\frac{\mathrm{d} }{\mathrm{d} x} a^x=?

A:不直接求 \frac{\mathrm{d} }{\mathrm{d} x} f(x),先求\frac{\mathrm{d}}{\mathrm{d} x} \ln_{}{f(x)}

\begin{aligned} \mu&=f(x)\\ \frac{\mathrm{d}}{\mathrm{d} x} \ln_{}{\mu } &=\frac{\mathrm{d} \ln_{}{\mu }} {\mathrm{d} \mu}\frac{\mathrm{d} \mu}{\mathrm{d} x} \\ &=\frac{1}{\mu}\frac{\mathrm{d} \mu}{\mathrm{d} x}\\ (\frac{\mathrm{d}}{\mathrm{d} x} \ln_{}{\mu } &=\frac{1}{\mu})\\ \\ \frac{\mathrm{d}}{\mathrm{d} x} \ln_{}{f(x)}&=\frac{f'}{f}\\ \ln_{}{a^x}&=x\ln_{}{a}\\ \frac{\mathrm{d} }{\mathrm{d} x} \ln_{}{a^x}&=\frac{\mathrm{d} }{\mathrm{d} x}x\ln_{}{a}\\ &=\ln_{}{a}\\ &=\frac{f'}{f}\\ \frac{\mathrm{d} }{\mathrm{d} x}a^x &=a^x\ln_{}{a} \end{aligned}

对数函数求导:

Q:\frac{\mathrm{d} }{\mathrm{d} x} \ln_{}{} x=?

A:\begin{aligned} w &=\ln x \\ e^{w}=& e^{\ln x}=x \\ \frac{d}{d x} e^{w} &=\frac{d}{d x} x=1 \\ \frac{d e^{w}}{d w} \frac{d w}{d x} &=e^{w} \frac{d w}{d x}=1 \\ \frac{d w}{d x} &=\frac{1}{e^{w}}=\frac{1}{x} \end{aligned}

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转载自blog.csdn.net/Dugege007/article/details/105472322
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