In American competitive exams such as the SAT, ACT, and AP Calculus, knowledge of the derivative of exponential functions is crucial in solving advanced mathematical problems. The derivative of an exponential function can be obtained using the definition of the derivative and incorporates the natural logarithm, a fundamental concept in calculus. This derivative is the same as the product of the original exponential function and the natural logarithm of its base, commonly called the derivative rule of exponential functions. In this article, we will discuss derivatives of exponential functions, such as key formulas, proofs, fractional derivatives, and graphs. We will also do practice problems to reinforce your learning of this fundamental concept.

Derivative of Exponential Function

Suppose \(a\) and \(x\) represent a constant and a variable respectively then the exponential function is written as \(a^{x}\) in mathematics. The derivative of \(a\) raised to the power of \(x\) with respect to \(x\) is written in the following form in calculus:

\(\frac{\mathrm{d}}{\mathrm{d}x}(a^{x})\)

The differentiation or derivative of the \(a^{x}\) function with respect to \(x\) is denoted as \(\frac{\mathrm{d}(a^{x})}{\mathrm{d}x}\) and is also simply written as \((a^{x})’\).

The derivative of \(a\) raised to the \(x\)-th power with respect to \(x\) is equal to the product of \(a\) to the \(x\)-th power and the natural logarithm of \(a\), i.e.

\(\frac{\mathrm{d}}{\mathrm{d}x}(a^{x})=a^{x}\log_{e}a\)

\(\because\) \(\ln=\log_{e}\)

\(\Rightarrow\) \(\frac{\mathrm{d}}{\mathrm{d}x}(a^{x})=a^{x}\log_{e}a=a^{x}\ln(a)\)

It is known as the differentiation rule of exponential function and it is used to find the derivative of any exponential function.

For example: Evaluate \(\frac{\mathrm{d}}{\mathrm{d}x}(2^{x})\)

Here the constant \(a=2\)

Now, substitute it in the differentiation rule of exponential function to find its derivative.

\(\frac{\mathrm{d}}{\mathrm{d}x}(2^{x})=2^{x}\log_{e}(2)=2^{x}\ln(2)\)

Hence, it can be used as a formula to find the differentiation of any function in exponential form.

Important points:

• One of the popular forms of the exponential function is \(f(x) = e^{x}\), where ‘\(e\)’ is “Euler’s number” and \(e = 2.718…\) approximately. The derivative of exponential function \(f(x) = e^{x}\) is given by \(f'(x) = e^{x}\).
• Exponential function \(f(x) = a^{x}\) is not defined for \(a<0\) and so does the derivative of exponential function.

Derivative of Exponential Function Formula

The formula for derivative of exponential function is given by:

• \(f(x)=a^{x}\), then \(f'(x)=a^{x}\log_{e}(a)=a^{x}\ln(a)\), or \(\frac{\mathrm{d}(a^{x})}{\mathrm{d}x}=a^{x}\log_{e}(a)=a^{x}\ln(a)\).
• \(f(x)=e^{x}\), then \(f'(x)=e^{x}\), or \(\frac{\mathrm{d}(e^{x})}{\mathrm{d}x}=e^{x}\).

Partial Derivative of Exponential Function:

In general when we’re given a function \(f(x)=e^{g(x)}\), then the derivative of the function is given as

\(f'(x)=e^{g(x)}\cdot g'(x)\).

Now how’s that changed when you have a function of two variables, and this is where partial derivatives come into picture. Let’s understand partial derivatives of exponential function with the help of examples.

Example 1: \(f(x,y)=e^{xy}\).

First find the first order partial derivative with respect to \(x\) (this means that \(y\) in the above function is treated as just a number), is given as

\(f_{x} = e^{xy}(y) = ye^{xy}\).

Now find the first order partial derivative with respect to \(y\) (this means that \(x\) in the above function is treated as just a number), is given as

\(f_{y} = e^{xy}(x) = xe^{xy}\).

Example 2: \(f(x,y)=e^{3x^{2}y}\).

\(f_{x}=e^{3x^{2}y}(3y)\frac{\mathrm{d}}{\mathrm{d}x}(x^{2})\)

\(\Rightarrow\) \(f_{x}=e^{3x^{2}y}(3y)(2x)\)

\(\Rightarrow\) \(f_{x}=e^{3x^{2}y}(3y)(2x)\)

\(\Rightarrow\) \(f_{x}=6(xy)e^{3x^{2}y}\).

\(f_{y}=e^{3x^{2}y}(3x^{2})\frac{\mathrm{d}}{\mathrm{d}y}(y)\)

\(\Rightarrow\) \(f_{y}=e^{3x^{2}y}(3x^{2})(1)\)

\(\Rightarrow\) \(f_{y}=3(x^{2})e^{3x^{2}y}\).

Let’s look at second order partial derivatives for this example.

First find the second order partial derivative with respect to \(x\),

\(f_{xx}=\frac{\mathrm{d}}{\mathrm{d}x}(f_{x})\)

\(\Rightarrow\) \(f_{xx}=\frac{\mathrm{d}}{\mathrm{d}x}(6(xy)e^{3x^{2}y})\)

Use the product rule to solve the above second order partial derivative, we get

\(\Rightarrow\) \(f_{xx}=6\left[e^{3x^{2}y}\frac{\mathrm{d}}{\mathrm{d}x}(xy)+(xy)\frac{\mathrm{d}}{\mathrm{d} x}(e^{3x^{2}y})\right]\)

\(\Rightarrow\) \(f_{xx}=6[e^{3x^{2}y}(y)+(xy)e^{3x^{2}y}(6xy)]\)

\(\Rightarrow\) \(f_{xx}=6ye^{3x^{2}y}[1+6x^{2}y]\)

Similarly, we can find the mixed second order partial derivative with respect to \(y\), i.e. \(f_{xy}=e^{3x^{2}y}(3x^{2})(6xy)+e^{3x^{2}y}(6x)\)

\(\Rightarrow\) \(f_{xy}=6xe^{3x^{2}y}[3x^{2}y+1]\)

Derivative of Exponential Function Proof using First Principle:

Now we prove that the derivative of exponential function \(a^{x}\) is \(a^{x}\ln(a)\) using the first principle of differentiation.

First write the derivative of this function in limit form by the definition of the derivative,

\(\frac{\mathrm{d}}{\mathrm{d}x}(a^{x})=\displaystyle \lim_{h\to 0}\frac{a^{x+h}-a^{x}}{h}\)

We know that, \(a^{m}\times a^{n}=a^{m+n}\)

\(\Rightarrow\) \(\frac{\mathrm{d}}{\mathrm{d}x}(a^{x})=\displaystyle \lim_{h\to 0}\frac{a^{x}\times a^{h}-a^{x}}{h}\)

\(\Rightarrow\) \(\frac{\mathrm{d}}{\mathrm{d}x}(a^{x})=\displaystyle \lim_{h\to 0}\frac{a^{x}(a^{h}-1)}{h}\)

\(\Rightarrow\) \(\frac{\mathrm{d}}{\mathrm{d}x}(a^{x})=a^{x}\displaystyle \lim_{h\to 0}\frac{a^{h}-1}{h}\)

We know that, \(\displaystyle \lim_{h\to 0}\frac{a^{h}-1}{h}=\ln(a)\)

\(\Rightarrow\) \(\frac{\mathrm{d}}{\mathrm{d}x}(a^{x})=a^{x}\ln(a)\)

Hence we have derived the derivative of exponential function using the first principle of derivatives.

Graph of Derivative of Exponential Function

The graph of the exponential function \(f(x) = b^{x}\) is increasing when \(b>1\) whereas \(f(x) = b^{x}\) is decreasing when \(b<1\). Thus, the graph of exponential function \(f(x) = b^{x}\)

• increases when \(b>1\)
• decreases when \(0<b<1\)

Since the graph of derivative of exponential function is the slope function of the tangent to the graph of the exponential function, therefore the graph of derivative of exponential function \(f(x) = b^{x}\) is increasing for \(b>1\). Given below is the graph of the exponential function and the graph of the derivative of the exponential function for \(b>1\) and \(0<b<1\).

Derivative of Exponential Function Solved Examples

1.Find the derivative of \(10^{x^{2}}\).

Solution: We know that \(\frac{\mathrm{d}}{\mathrm{d}x}(a^{x})=a^{x}\log_{e}a=a^{x}\ln(a)\)

Here \(a = 10\) and \(x = x^{2}\)

\(\frac{\mathrm{d}}{\mathrm{d}x}(10^{x^{2}})=10^{x^{2}}\log_{e}(10)\cdot \frac{\mathrm{d}}{\mathrm{d}x}(x^{2})\)

\(\Rightarrow\) \(\frac{\mathrm{d}}{\mathrm{d}x}(10^{x^{2}})=10^{x^{2}}\log_{e}(10)\cdot (2x)\)

\(\Rightarrow\) \(\frac{\mathrm{d}}{\mathrm{d}x}(10^{x^{2}})=10^{x^{2}}\log_{e}(10)(2x)=10^{x^{2}}\ln(10)(2x)\)

Therefore, the derivative of \(10^{x^{2}}\) is \(10^{x^{2}}\log_{e}(10)(2x)\) or \(10^{x^{2}}\ln(10)(2x)\).

2.Find the derivative of \(3^{x}+3x^{2}\).

Solution: We know that \(\frac{\mathrm{d}}{\mathrm{d}x}(a^{x})=a^{x}\log_{e}a=a^{x}\ln(a)\)

\(\frac{\mathrm{d}}{\mathrm{d}x}(3^{x}+3x^{2})=\frac{\mathrm{d}}{\mathrm{d}x}(3^{x})+\frac{\mathrm{d}}{\mathrm{d}x}(3x^{2})\)

\(\Rightarrow\) \(\frac{\mathrm{d}}{\mathrm{d}x}(3^{x}+3x^{2})=3^{x}\ln(3)+6x\)

Therefore, the derivative of \(3^{x}+3x^{2}\) is \(3^{x}\ln(3)+6x\).

3.Find the derivative of \(\sin(e^{3x})\).

Solution: We know that \(\frac{\mathrm{d}}{\mathrm{d}x}(a^{x})=a^{x}\log_{e}a=a^{x}\ln(a)\)

Let \(y = \sin(u)\), where \(u = e^{3x}\)

\(\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}u}\frac{\mathrm{d}u}{\mathrm{d}x}\)

\(\Rightarrow\) \(\frac{\mathrm{d}y}{\mathrm{d}x}=(\cos(u))(3e^{3x})\)

Substitute \(u = e^{3x}\), we get

\(\Rightarrow\) \(\frac{\mathrm{d}y}{\mathrm{d}x}=(\cos(e^{3x}))(3e^{3x})\)

\(\Rightarrow\) \(\frac{\mathrm{d}y}{\mathrm{d}x}=3e^{3x}\cos(e^{3x})\)

Therefore the derivative of \(\sin(e^{3x})\) is \(3e^{3x}\cos(e^{3x})\).

We hope this article has been helpful in enhancing your understanding and preparing you for U.S. competitive exams like the SAT, ACT, and AP Calculus. Stay connected with reliable resources for updates on related math topics and other essential subjects. Additionally, consider exploring practice tests and test series to assess your knowledge and improve your performance on exam day.