2.7 Derivatives of cos x, sin x, e^x, and ln x

Now that you’ve learned how to find derivatives of polynomial equations, it’s time to learn how to find derivatives of special functions including $\sin x$, $\cos x$, $e^x$, and $\ln x$. Finding these derivatives are relatively simple as long as you can remember the rules. 👍

😎 Derivatives of Special Functions

Before we get into each individual rule, here’s a quick table summarizing them.

Derivative of $\sin x$

The derivative of $\sin x$ will always be $\cos x$. Let’s look at an example:

When finding the derivative of this equation, we need to find the derivative of $4\sin x$ and $3x$ separately.

Since the derivative of $\sin x = \cos x$, the derivative of the first part of the equation is $4\cos x$. The derivative of $3x$ is $3$. Therefore $f'(x) = 4cosx+3$.

Derivative of $\cos x$

The derivative of $\cos x$ will always be $-\sin x$. Let’s look at an example:

When finding the derivative of this equation, we need to find the derivative of $2\cos x$ and $3$ separately.

To find the derivative of $2\cos x$, we need to know that the derivative of $\cos x$ is $-\sin x$. Therefore, the derivative of the first part of the equation is $-2\sin x$. The derivative of 3 is 0, as explained in an earlier lesson discussing the constant rule. Therefore the derivative of the above equation is $-2\sin x$.

Derivative of $e^x$

This one is pretty straightforward. The derivative of $e^x$ is simply… $e^x$! That’s right, the derivative of $e^x$ is just itself. 🤯

Here’s an example:

The derivative of the first part of the equation is $e^x$, since we just stated that the derivative of $e^x$ is itself. The derivate of the second part of the equation is $12x^3$, according to the power rule. Therefore, $f'(x) = e^x + 12x^3$.

Derivative of $\ln x$

The derivative of $\ln x$ is $\frac {1}{x}$. Let’s look at an example:

The derivative of the first part of the equation is $\frac {5}{x}$ since we know that the derivative of $\ln x$ is $\frac {1}{x}$. The derivative of $2x$ = $2$, so $f'(x)=\frac {5}{x}+2$.

These rules take a little bit of practice, but once you memorize them, it gets simpler! You got this. 🍀