# Math Insight

### Derivative of polynomials

Math 201, Spring 2017
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Total points: 3
1. Calculate the derivatives of the following power functions.
1. $f(x)=x^3$:     $f'(x)=$
2. $g(x)=x^5$:     $\diff{g}{x} =$
3. $g(y) = y^n$, for any number $n$:     $\diff{g}{y}=$
4. $f(z) = z^m$ for any number $m$:     $f'(z)=$
5. $g(x) = 1$ (You can think of $g(x)=x^0$.):     $g'(x) =$
6. $f(y) = \sqrt{y}$:     $f'(y) =$
7. $\displaystyle f(x) = \frac{1}{x}$:     $\displaystyle \diff{f}{x} =$
8. $\displaystyle g(y) = \frac{1}{y^2}$:     $g'(y) =$
9. $h(t)=t^{2/3}$:     $\diff{h}{t}=$

2. The derivative of a polynomial is based on the derivative of power functions plus some basic properties of the derivative.
1. Derivative of a constant multiple. Let $f(x)$ be a differentiable function with derivative $\diff{f}{x}$. Let $g(x)=3f(x)$ and let $h(x)=cf(x)$ for some constant number $c$. What are the derivatives $\diff{g}{x}$ and $\diff{h}{x}$?
$\diff{g}{x}=$

$\diff{h}{x} =$
2. If $f(x)=x^3$, what is $\diff{f}{x}$?

If $g(x)=4f(x)=4x^3$, what is $\diff{g}{x}$?

If $h(x)=cf(x)=cx^3$, what is $\diff{h}{x}$?
3. Derivative of constant. What is the derivative of $f(x)=c$ for a constant $c$?
4. Derivative of a function plus a constant. Let $f(x)$ be some differentiable function with derivative $\diff{f}{x}$. Let $g(x)=f(x) + c$ for some constant number $c$. What is the derivative of $g$? $\diff{g}{x} =$
5. Derivative of a sum. Let $f(x)$ be a differentiable function with derivative $\diff{f}{x}$. Let $g(x)$ be another differentiable function with derivative $\diff{g}{x}$. Let $h(x)$ be the sum of $f$ and $g$, $h(x)=f(x)+g(x)$. What is the derivative of $h$?
$\diff{h}{x}=$
6. If $f(x)=x^2$ and $g(x)=x$, what are $\diff{f}{x}$ and $\diff{g}{x}$?
$\diff{f}{x}=$

$\diff{g}{x}=$

If $h(x)=f(x)+g(x)=x^2+x$, what is $\diff{h}{x}$?

7. If $f(x)=10x^3$ and $g(x)=5$, what are $\diff{f}{x}$ and $\diff{g}{x}$?
$\diff{f}{x}=$

$\diff{g}{x}=$

If $h(x)=f(x)+g(x)=10x^3+5$, what is $\diff{h}{x}$?

8. Let $g(x)=11x^5$ and $h(x)=8x^9$. What are $g'(x)$ and $h'(x)$?
$g'(x)=$

$h'(x)=$

Let $f(x)=g(x)+h(x)$. What is $f'(x)$?

3. Calculate derivatives of the following polynomials.
1. Find $\diff{p}{x}$ for $p(x) = 17 x^{3} - 5 x + 11$. $\diff{p}{x}=$

Calculate $\diff{p}{x}\bigr|_{x=-1}=$
2. Find $q'(x)$ for $q(x) = -3x^{11}+\frac{3}{4}x^8-\frac{1}{3}x^3$.
$q'(x)=$
. Calculate $q'(2)$ =
.
3. Find $\displaystyle\diff{}{x} \Bigl(\frac{x^{3}}{6} + \frac{x^{2}}{2} + x + 1 \Bigr)$ =
.

4. Let $f(z)=n + 6 z^{n}$ for some number $n$. Calculate $\diff{f}{z}$ =

5. Let $g(y)=- c^{2} + c y^{5} + \frac{c y^{2}}{2}$ for some number $c$. Calculate:
$g'(y)=$

$g'(1)=$

6. Let $f(x)=- 6 x^{3} - x$. Calculate:
$f'(x)=$

$f'(2)=$

$f'(a)=$

$f'(\heartsuit) =$