Special derivatives

What you’ll learn

• Derivatives of exponential, logarithmic, and trigonometric functions

So far, we’ve worked with derivatives of polynomials and rational functions. This section introduces the derivatives of non-algebraic functions.

Exponential functions

Exponential functions model rapid growth and decay and show up in population models, radioactive decay, and compound interest.

The most important exponential function is the natural exponential function $e^x$, whose rate of change is equal to itself.

Derivative of the natural exponential function:

$$\frac{d}{dx}(e^x)=e^x$$

More generally, for an exponential function of the form $f(x)=b^x$ (where $b>0$ and $b\ne 1$), the derivative is

Derivative of a general exponential function:

$$\frac{d}{dx}(b^x)=b^x\cdot \ln (b)$$

In other words, rewrite the original function and multiply by $\ln (b)$.

Sidenote

Power rule vs. derivatives of exponentials

Be careful not to mix these up. Here, $x$ is a variable and $b$ is a constant.

1. Power rule if $f(x)=x^b$:

   • e.g. $\frac{d}{dx}(x^3)=3x^2$

2. Rule for exponentials if $f(x)=b^x$:

   • e.g. $\frac{d}{dx}(3^x)=3^x\ln (3)$

Examples

1. At what value(s) of $x$ does the graph of

$$f(x)=x^2{e}^x$$

have horizontal tangent lines?

2. At what value of $x$ does the graph of

$$f(x)=\frac{x}{7^x}$$

have a horizontal tangent line?

Logarithmic functions

Derivative of the natural logarithmic function:

$$\frac{d}{dx}(\ln x)=\frac{1}{x}$$

More generally, any logarithmic function $f(x)={\log }_{b}x$, where the base $b$ is a positive number and $b\ne 1$, can be rewritten using the change-of-base rule:

$$\begin{array}{rl}f(x)& =\frac{\ln x}{\ln (b)}\\ & =\frac{1}{\ln (b)}\cdot \ln x\end{array}$$

Then, since $\frac{1}{\ln b}$ is a constant, it stays out front when differentiating.

Derivative of a general logarithmic function:

$$\frac{d}{dx}{\log }_{b}x=\frac{1}{x\ln (b)}$$

Sidenote

The domain of the derivative

A derivative formula is only valid where the original function is defined. For logarithms, the argument must be $>0$.

Examples

1. Determine the point at which the tangent line to $f(x)=x\ln (x)$ is parallel to the line $y=3x-1$.

2. Let $f(x)={\log }_{k}x$, where $k$ is a positive real number. Find the value of $k$ such that the slope of the tangent line at $x=1$ equals $2$.

Trigonometric functions

Begin with the basic trig functions $\sin x$ and $\cos x$.

$$\begin{array}{rl}\frac{d}{dx}(\sin x)& =\cos x\\ \frac{d}{dx}(\cos x)& =-\sin x\end{array}$$

From there, the other trig functions can be differentiated by first rewriting in terms of $\sin x$ and $\cos x$.

For example, $f(x)=\tan x$ can be rewritten as the quotient

$$\tan x=\frac{\sin x}{\cos x}$$

Then differentiating using the quotient rule,

$$\begin{array}{rl}{f}^{\prime }(x)& =\frac{\cos x\cdot \cos x-\sin x\cdot (-\sin x)}{{\cos }^2(x)}\\ & =\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}\end{array}$$

Using the Pythagorean identity ${\sin }^{2}x+{\cos }^{2}x=1$,

$$f^{\prime }(x)=\frac{1}{{\cos }^{2}x}=\boxed{{\sec }^{2}x}$$

The same can be done to $f(x)=\cot x$.

Find the derivative of $f(x)=\cot x$ by rewriting it as a quotient.

Now do the same for the remaining two reciprocal functions:

Find the derivatives of

a) $f(x)=\sec x$

b) $f(x)=\csc x$

Solutions

a) Differentiate $f(x)=\sec x$

b) Differentiate $f(x)=\csc x$

Memorizing the trig derivatives saves time, but it’s also useful to know how to derive them from the derivatives of $\sin x$ and $\cos x$.

Higher-order derivatives

Because a derivative is just another function, you can differentiate it again to find a second derivative, and repeat the process indefinitely through repeated differentiation. For example, if $f(x)=x^2$, then its first 3 derivatives are

$$\begin{array}{rl}{f}^{\prime }(x)& =2x\\ f^{\prime \prime }(x)& =2\\ f^{\prime \prime \prime }(x)& =0\end{array}$$

While polynomials eventually differentiate down to zero, other functions exhibit different patterns. For example, trigonometric functions like $\sin x$ and $\cos x$ repeat in a four-step cycle, allowing them to be differentiated indefinitely.

These higher-order derivatives are often denoted $f^{(n)}(x)$. where $n$ indicates the number of times the function has been differentiated.

Note: In AP Calculus AB, applications are generally limited to the 1st and 2nd derivatives.