derivative of bases other than e | ap calculus bc,12th grade math

# Derivative of bases other than e

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Derivative of bases other than e : To differentiate exponential and logarithmic functions to other bases, you have three options:
(1) use the definitions of $a^{x} and \log_{a}{x}$ and differentiate using the rules for the natural exponential and logarithmic functions.
(2) use logarithmic differentiation, or
(3) use the differentiation rules for bases other than given in the theorem below.

Let 'a' be any positive real number and 'u' be a differentiable function of x.
1) $\frac{\text{d}}{\text{d}x}(a^{x}) = a^{x}. ln(a)$

2) $\frac{\text{d}}{\text{d}x}(a^{u}) = a^{u}. ln(a).\frac{\text{d}u}{\text{d}x}$

2) $\frac{\text{d}}{\text{d}x}(\log_{a}{x})= \frac{1}{ln(a).x}$

3) $\frac{\text{d}}{\text{d}x}(\log_{a}{u})= \frac{1}{ln(a).u}\frac{\text{d}u}{\text{d}x}$