##
Techniques of Differentiation

Maybe the
easiest and most useful formulas are the ones that say that the
derivative is linear:

Combined with the formula
(*x*^{n})' = *n x*^{n-1}, we see that every
polynomial function has a derivative at any point.
**Example.** For
*P*(*x*) = 1-2*x* + 3*x*^{4} -5 *x*^{6}, we have

The next two formulas are the most powerful ones. They deal with
the derivative of a product and a quotient. They are commonly
called the **product rule** and the **quotient rule**. We have

In particular, we have

So, we have

which means that the formula
(*x*^{r})' = *r x*^{r-1} is also valid
for negative exponents.
Before we discuss the derivative of trigonometric functions, let
us stop here and reflect a little bit more on polynomial
functions. Indeed, we saw that the derivative of a polynomial
function is also a polynomial function. So we can take another
derivative and generate a new function. This function is called
**the second derivative**. We can keep doing this as long as we
want to. The functions obtained are called higher derivatives.
The common notations used for them are

**Exercise 1.** Find the derivative of the function

Is there a nice way to rewrite this derivative?
**Answer.**

**Exercise 2.** Find the derivative of

**Answer.**

**Exercise 3.** Solve the equation
when

**Answer.**

**Exercise 4.** Find the points on the graph of
*y* = *x*^{3/2} -
*x*^{1/2} at which the tangent line is parallel to the line *y*+2*x*
= 1. Also find the points on the same graph at which the
tangent line is perpendicular to the line *y*-*x* = 3.

**Answer.**

**
**

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