Chapter 2: Differentiation

Section 2.6: Derivatives of the Exponential and Logarithmic Functions

Example 2.6.6

$$ Use logarithmic differentiation to obtain the derivative of $y\left(x\right)\=a^x$. $$ Solution $$ Take the logarithm of both sides of $y\left(x\right)\=a^x$ and differentiate, as per the following calculations. $$ $y\left(x\right)$ $\= a^x$ $\ln \left(y\left(x\right)\right)$ $\= \ln \left(a^x\right)$ $\frac{d}{\mathrm{dx}} \ln \left(y\left(x\right)\right)$ $\= \frac{d}{\mathrm{dx}}\left(x \ln \left(a\right)\right)$ $\frac{y\prime \left(x\right)}{y\left(x\right)}$ $\= \ln \left(a\right)$ $y\prime \left(x\right)$ $\= a^x \ln \left(a\right)$ $$

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