3.5: Derivatives of Exponential and Hyperbolic Functions (2024)

Discovering \(e\)

In calculus (and often in precalculus), acommon "beginning" limit to investigate numerically is

\[ \displaystyle\lim_{k \to \infty}{\left(1 + \frac{1}{k}\right)^k}. \nonumber \]

Table \(\PageIndex{1}\) shows values of this limit for large values of \(k\).

We went a little overboard, but you get the point. Itappearsthis limit is approaching some value around \(2.71828\ldots\). This very special value is a non-terminating, non-repeating number. That is, it is an irrational number - justlike \(\pi\). In fact, if we could assign an importanceto the value of this limit, it would be tied with the importance of the numbers \(0\), \(1\), \(\pi\), and \(i = \sqrt{-1}\).

This limit comes up so often in the sciences that we give it a special name. Rather than calling it, "the limit that approaches \(2.718281828459045\ldots\)," we call it \(e\). As was mentioned, itappearsvia our table that this limit approaches a constant (that we are deciding to label\(e\)); however, the proof of this fact is reserved for later in the course. For now, it's important to have this limit in hand for use.

An important alternative form for this limit is found by letting \(n = \frac{1}{k}\). As \(k \to \infty\), \(n \to 0\) and we get the form

\[ \displaystyle\lim_{n\to 0}{\left(1 + n\right)^{1/n}}. \nonumber \]

Before we move on, let's formalize this definition.

Fear not, we will be using this special number shortly; however, before we do, we need to discuss some Calculus.

Derivatives Exponential Functions

Just as when we found the derivatives of other functions, we can find the derivatives of exponential functions using formulas. Recall that all non-transformed exponential functions have the basic form \(B(x) = b^x\), where \(b \gt 0\) and \(b \neq 1\). Our initial goal is to prove that exponential functions are continuous everywhere. Once we have this, the proof of the derivative of an exponential function becomes trivial.

Derivatives of the Hyperbolic Functions

Recall that the hyperbolic sine and hyperbolic cosine are defined as

\[\sinh x=\dfrac{e^x−e^{−x}}{2} \nonumber \]

and

\[\cosh x=\dfrac{e^x+e^{−x}}{2}. \nonumber \]

The other hyperbolic functions are then defined in terms of \(\sinh x\) and \(\cosh x\). The graphs of the hyperbolic functions are shown in Figure \(\PageIndex{1}\).

It is easy to develop differentiation formulas for the hyperbolic functions. For example, looking at \(\sinh x\) we have

\[\begin{array}{rcl}
\dfrac{d}{dx} \left(\sinh x \right) & = & \dfrac{d}{dx} \left(\dfrac{e^x−e^{−x}}{2}\right) \\
& = & \dfrac{1}{2}\left[\dfrac{d}{dx}(e^x)−\dfrac{d}{dx}(e^{−x})\right] \\
& = & \dfrac{1}{2}[e^x+e^{−x}] \\
& = \cosh x. \\
\end{array} \nonumber \]

Similarly,

\[\dfrac{d}{dx} \cosh x=\sinh x. \nonumber \]

We summarize the differentiation formulas for the hyperbolic functions in Table \(\PageIndex{2}\).

Let’s take a moment to compare the derivatives of the hyperbolic functions with the derivatives of the standard trigonometric functions. There are a lot of similarities, but differences as well. For example, the derivatives of the sine functions match:

\[\dfrac{d}{dx} \sin x=\cos x \nonumber \]

and

\[\dfrac{d}{dx} \sinh x=\cosh x. \nonumber \]

The derivatives of the cosine functions, however, differ in sign:

\[\dfrac{d}{dx} \cos x=−\sin x, \nonumber \]

but

\[\dfrac{d}{dx} \cosh x=\sinh x. \nonumber \]

Key Concepts

• On the basis of the assumption that the exponential function \(y=b^x, \,b>0\) is continuous everywhere and differentiable at \(0\), this function is differentiable everywhere and there is a formula for its derivative.
• We can use a formula to find the derivative of \(y=\ln x\), and the relationship \(\log_b x=\dfrac{\ln x}{\ln b}\) allows us to extend our differentiation formulas to include logarithms with arbitrary bases.
• Logarithmic differentiation allows us to differentiate functions of the form \(y=g(x)^{f(x)}\) or very complex functions by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating.
• Hyperbolic functions are defined in terms of exponential functions.
• Term-by-term differentiation yields differentiation formulas for the hyperbolic functions. These differentiation formulas give rise, in turn, to integration formulas.
• With appropriate range restrictions, the hyperbolic functions all have inverses.
• Implicit differentiation yields differentiation formulas for the inverse hyperbolic functions, which in turn give rise to integration formulas.
• The most common physical applications of hyperbolic functions are calculations involving catenaries.

Key Equations

• Derivative of the natural exponential function

\(\dfrac{d}{dx}\Big(e^{g(x)}\Big)=e^{g(x)}g′(x)\)

• Derivative of the natural logarithmic function

\(\dfrac{d}{dx}\Big(\ln g(x)\Big)=\dfrac{1}{g(x)}g′(x)\)

• Derivative of the general exponential function

\(\dfrac{d}{dx}\Big(b^{g(x)}\Big)=b^{g(x)}g′(x)\ln b\)

• Derivative of the general logarithmic function

\(\dfrac{d}{dx}\Big(\log_b g(x)\Big)=\dfrac{g′(x)}{g(x)\ln b}\)

Glossary

catenary

a curve in the shape of the function \(y=a\cdot\cosh(x/a)\) is a catenary; a cable of uniform density suspended between two supports assumes the shape of a catenary

logarithmic differentiation

is a technique that allows us to differentiate a function by first taking the natural logarithm of both sides of an equation, applying properties of logarithms to simplify the equation, and differentiating implicitly