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

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.

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.

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

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.

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

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

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

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

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

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.

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

Latest Math Topics

Latest Math Problems

Email subscription

Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers.
Know more

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.