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Phone 435-797-1494
Contact Christian Bolander
Email christian.bolander@usu.edu

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Derivatives of Exponential and Logarithmic Functions

Engineering Context:

MAE: The function of a decreasing temperature is an exponential function. It is important for mechanical engineers to understand derivatives of exponential functions so that they can analyze the temperature changes of a system. This would help them design efficient systems.

T_{s, 2} + ∆ T \frac{ln (r/ r_{2})}{ln (r_{1}/ r_{2})}

ECE: To find the relationship between voltage in and voltage out, electrical engineers use the following equation. To find the change in voltage, these engineers would need to know how to take the derivative of this function.

V_{out} = K ln (\frac{V_{in}}{V_{ref}})

CEE/BE: Environmental and bilogical engineers solve problems about bacteria growth. These calculations often involve the exponential growth formula $P = P_{0} e^{r t}$ . Finding the rate of change of the population of bacteria would require taking the derivative of this exponential function.

The Essentials

Derivative of Exponential Function:

\frac{d}{d x} e^{f (x)} = e^{f (x)} \cdot f^{'} (x)

Derivative of a Constant Exponential Function:

$\frac{d}{d x} a^{x} = a^{x} ln (a)$ , where $a$ is some constant

Derivative of a General Logarithm:

\frac{d}{d x} {log}_{b} (g (x)) = \frac{1}{g (x) ln (b)} \cdot g^{'} (x)

Derivative of a Natural Logarithm:

\frac{d}{d x} ln (g (x)) = \frac{1}{g (x)} \cdot g^{'} (x)

A Deeper Dive

Understanding why each of these derivatives are valid is quite interesting. Below are some proofs of the derivatives of logarithmic and exponential functions. Exponential functions and natural logarithms are inverse functions of one another. This is helpful to keep in mind when looking at these proofs.

Proof of the Derivative of an Exponential

y = e^{x}

Use the definition of the derivative:

\frac{d y}{d x} = lim_{h \to 0} \frac{e^{x + h} - e^{x}}{h} = \frac{e^{x} (e^{h}- 1)}{h}

\frac{d y}{d x} = e^{x} lim_{h \to 0} \frac{e^{h} - 1}{h}

\frac{d y}{d x} = e^{x} \cdot 1 = e^{x}

Proof of the Derivative of a General Logarithm:

{log}_{b} (x) = y \to b^{y} = x

Take the natural log of both sides.

ln (b^{y}) = ln (x)

Use log power rules.

y ln (b) = ln (x)

Isolate y.

y = \frac{ln (x)}{ln (b)}

Take the derivative with respect to x.

\frac{d y}{d x} = \frac{1}{ln (b)} \cdot \frac{1}{x} = \frac{1}{x ln (b)}

Proof of the Derivative of a Natural Logarithm:

y = ln (x) \to e^{y} = e^{ln (x)} \to e^{y} = x

Use implicit differentiation.

e^{y} \frac{d y}{d x} = 1

\frac{d y}{d x} = \frac{1}{e^{y}}

\frac{d y}{d x} = \frac{1}{x}

Practice

Find the following derivatives:

Exercise 1. $\frac{d}{d x} ln (5 x^{2})$

Exercise 2. $\frac{d}{d x} e^{2 x^{3} + 3 x}$

Solutions:

Exercise 1.

\frac{d}{d x} ln (5 x^{2}) = \frac{1}{5 x^{2}} \cdot 10 x = \frac{2}{x}

Exercise 2.

\frac{d}{d x} e^{2 x^{3} + 3 x} = e^{2 x^{3} + 3 x} \cdot (6 x^{2} + 3) = (6 x^{2} + 3) e^{2 x^{3} + 3 x}

Derivatives of Exponential and Logarithmic Functions

Engineering Context:

The Essentials

A Deeper Dive

Practice

More Resources