Math Doubts

Proof of $\dfrac{d}{dx} a^x$ formula

$\large \dfrac{d}{dx}{a^x} \,=\, a^x \log_{e}{a}$

It is called differentiation of $a$ raised to the power of $x$ with respect to $x$ formula and this differentiation rule can be derived in differential calculus on the basis of relation between limit and differentiation. Here the steps to prove the $\dfrac{d}{dx}{a^x}$ formula mathematically.

Express differentiation of function in Limit form

The derivative of a function with respect to $x$ can be expressed in limit form as per the mathematical relation between limit and differentiation. So, the differentiation of $a$ raised to the power of $x$ is expressed in limit form.

$\implies$ $\dfrac{d}{dx}{a^x} = \displaystyle \lim_{h \,\to\, 0}{\dfrac{a^{x+h}-a^x}{h}}$

Split the exponential function

An exponential function in numerator contains sum of two literals as its exponent. It can be divided as two multiplying factors by applying product rule of exponents.

$\implies$ $\dfrac{d}{dx}{a^x} = \displaystyle \lim_{h \,\to\, 0}{\dfrac{a^{x} \times a^{h}-a^x}{h}}$

Simplifying the equation

$\implies$ $\dfrac{d}{dx}{a^x} = \displaystyle \lim_{h \,\to\, 0}{\dfrac{a^{x} \times (a^{h}-1)}{h}}$

$\implies$ $\dfrac{d}{dx}{a^x} = \displaystyle \lim_{h \,\to\, 0}{a^{x} \times \dfrac{a^{h}-1}{h}}$

The multiplying factor $a^x$ is a constant in this case. So, it can be taken out from the expression reasonably.

$\implies$ $\dfrac{d}{dx}{a^x} = a^{x} \times \displaystyle \lim_{h \,\to\, 0}{\dfrac{a^{h}-1}{h}}$

Obtaining the Result

According to limit rules, the lim of $\dfrac{a^x-1}{x}$ as $x$ approaches $0$ is equal to natural logarithm of $a$.

$\displaystyle \lim_{h \,\to\, 0}{\dfrac{a^{h}-1}{h}} = \log_{e}{a}$

Now simplify the equation for deriving the derivative of $a^x$ with respect to $x$ formula in differential calculus.

$\implies$ $\dfrac{d}{dx}{a^x} = a^{x} \times \log_{e}{a}$

$\,\,\, \therefore \,\,\,\,\,\,$ $\dfrac{d}{dx}{a^x} = a^{x} \log_{e}{a}$

It is used as formula to deal differentiation of $a^x$ with respect to $x$ in calculus. It can be simply written in natural logarithmic form as follows.

$\dfrac{d}{dx}{a^x} = a^{x} \ln{a}$

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