# Techniques to Evaluate Derivatives

## Basic Techniques

Last time, we talked about some of the different properties of derivatives and the limit definition of the derivative. As another example, lets try evaluating the derivative of $f(x)=x^5$. Using the limit definition and also expanding out the binomial term, we have

Simplifying gives

As we can see, this process is doable, but tedious and algebraically hairy. Let’s develop a set of rules to make finding derivatives a little easier:

- If $f(x)=\text{ constant}$, then $f’(x)=0$.
- If $f(x)=mx+b$, then $f’(x)=m$.
**Power Rule**: If $f(x)=x^n$, then $f’(x)=nx^{n-1}$. Note that $n$ does not have to be an integer; it can be any real number!**Linearity**: If $h(x)=\alpha f(x)\pm \beta g(x)$ for constants $\alpha, \beta$, then $h’(x)=\alpha f’(x)\pm \beta g’(x)$.**Product Rule**: If $h(x)=f(x)g(x)$, then $h(x)=f’(x)g(x)+g’(x)f(x)$.**Quotient Rule**: If $h(x)=f(x)/g(x)$, then

These are the basic rules. Of course, there are more advanced techniques possible to evaluate derivatives, but we’ll introduce them in the future. Let’s try a few examples:

## Example 1

Let’s evaluate the derivative of $f(x)=(x+1)^2$ using the power rule and linearity. We can expand $f(x)$ as $f(x)=x^2+2x+1$. The derivative of $x^2$ is $2x$, the derivative of $x$ is $1$, and the derivative of $1$ is $0$ using the power rule. Therefore, we have

## Example 2

Let’s evaluate the derivative of $f(x)=x^2=x\cdot x$ using the product rule. Here $f(x)=x$ (meaning $f’(x)=1$) and $g(x)=x$ (meaning $g’(x)=1$). Plugging this into the formula from above, we find

We can also evaluate this derivative using the power rule, where $n=2$:

Notice how we get the same result using either method of finding the derivative.

## Example 3

Let’s evaluate the derivative of $f(x)=1/x$ using the quotient rule. Here $f(x)=1$ (meaning $f’(x)=0$) and $g(x)=x$ (meaning $g’(x)=1$). Plugging this into the formula from above, we find

We can also evaluate this derivative using the power rule, where $n=-1$:

Notice how we get the same result using either method of finding the derivative.

## Exercises

### Problem 1

Evaluate the derivatives of the following functions:

- $f(x)=4x^5-5x^4$
- $f(x)=\frac{x}{1+x^2}$
- $f(x)=\frac{x^2-1}{x}$
- $f(x)=(3x^2)\sqrt{x}$
- $f(x)=2x-\frac{4}{\sqrt{x}}$
- $f(x)=(x^2+3)(x^3+4)$
- $f(x)=\frac{\frac{1}{x}+\frac{1}{x^2}}{x-1}$
*Hint: It may be easier to rewrite this function first.*

### Problem 2

Assume $f$ and $g$ are differentiable functions such that $f(2)=3$, $f’(2)=-1$, $f’(3)=7$, $g(2)=-5$, and $g’(2)=2$. Find the numerical value of the following expressions:

- $(g-f)’(2)$
- $(fg)’(2)$
- $(\frac{f}{g})’(2)$
- $(5f+3g)’(2)$