Deriving the Derivative of \(a^x\)

Derivation of \(\dfrac{\mathrm d}{\mathrm dx}(a^x)=a^x\ln a\)

The three common forms are \(a^x\), where the base is constant; \(x^n\), where the exponent is constant; and \(x^x\), where both the base and exponent vary.

To prove the derivative rule for \(a^x\), let \(y=a^x\), where \(a>0\) is constant. Taking natural logarithms gives \(\ln y=x\ln a\).

Differentiating implicitly with respect to \(x\) gives \(\frac1y\frac{\mathrm dy}{\mathrm dx}=\ln a\).

Hence \(\frac{\mathrm dy}{\mathrm dx}=y\ln a=a^x\ln a\), so \(\frac{\mathrm d}{\mathrm dx}(a^x)=a^x\ln a\).