
We review differentiation and integration.

### Review of derivative rules

One of the fundamental objects of differential calculus is the derivative. As a reminder, here are some important results:

Using the above rules only, what is the derivative of $\frac {1}{\sqrt [4]{x^3}}$ with respect to $x$?

Once the derivatives of basic functions have been established, we can use them to build the derivatives of more complicated functions:

What is the derivative of $4x^2 +\frac {2}{3\sqrt {x}}$ with respect to $x$?
What is the derivative of $x^2 \sin (x)$ with respect to $x$?

### Derivatives and Tangent Lines

While there are many useful applications of derivatives, one of the most important ones is the relationship between derivatives and tangent lines.

Suppose that $f(x)$ is a differentiable function on an interval $I$. If we pick a value $a$ in the interval, and “zoom in” on the graph around $x=a$, notice what happens:

As evidenced by the image, when the function $f(x)$ is differentiable at a given $x$-value, the graph of $f(x)$ becomes closer to a line as we “zoom in,” and we call this line the tangent line at $x=a$.

To find the equation of this line, we need a point of the line and the slope of the line. The slope of the line is $m_{tan} =f'(a)$ and the point on the line is $(a,f(a))$.

This result is also a direct consequence of the limit definition of the derivative and some algebra. If $f'(a)$ exists, then:

In order to write everything as a single limit, note we can write $f'(a)$ as:

We now return to the limit definition of $f'(a)$ and write:

Denoting $f'(a)(x-a) + f(a)$ by $l_a(x)$, we note that $l_a(x)= f'(a)(x-a) + f(a)$ is the equation of a line, and we can write the last equation above as:

Since this limit is $0$, we can interpret it geometrically by saying that as the distance between $x$ and $a$ becomes smaller, the distance between the function $f(x)$ and the line $l_a(x)$ becomes even smaller. That is, as we “zoom in” on the graph of a function at a point where it is differentiable, the graph becomes closer and closer to a line and we call this line the tangent line.