Differentiation rules > Product rule

Exercise 1

Find the derivative of the following functions.

a

$f\left(x\right)=({x}^{3}+6)(4{x}^{2}-5x)$

b

$g\left(x\right)=(10-x)\cdot \sqrt{x}$

c

$R\left(t\right)=3t{(t+5)}^{4}$

d

$y\left(x\right)=x\cdot \sqrt{5+{x}^{2}}$

e

$y\left(x\right)=x-\sqrt{5+{x}^{2}}$

f

$V\left(r\right)=(100-\frac{5}{r}){(20-r)}^{2}$

Exercise 2

Here you see the graphs of the functions ${y}_{1}\left(x\right)={x}^{2}$ and ${y}_{2}\left(x\right)={(2x-8)}^{4}$. The function $f\left(x\right)={y}_{1}\left(x\right)\cdot {y}_{2}\left(x\right)$ is the product function of both.

a

The zeroes of $f$ can be deduced from the graph. What are the zeroes of the graph of $f$?

b

Show that $f\prime \left(x\right)={(2x-8)}^{3}(12{x}^{2}-16x)$

c

Use the derivative to find th extrema of $f$.

d

For what values of $k$ does the equation $f\left(x\right)=k$ have exactly four solutions?

Exercise 3

Given is the function $f\left(x\right)=4x\sqrt{x}\cdot {(1-x)}^{3}$.

a

For what values of $x$ does the graph have a tangent parallell to the $x$ -axis?

b

This function has two extrema. What are they?

Exercise 4

Here you see the graph of the function $f\left(x\right)=x\cdot \sqrt{8-{x}^{2}}$ the way the graphing calculator depicts it.

a

The graph is incomplete. You can tell this from the zeroes of this function. What zeroes does the graph of $f$ have?

b

Calculate the range of $f$ using differentiation .

c

Compose the equation for the tangent to the graph of $f$ in $(0,0)$.

Exercise 5

Given is the function $f\left(x\right)=\mathrm{0,25}{x}^{2}-x\sqrt{x}$.

a

Algebraically calculate the range of $f$.

b

Calculate the coordinates of the inflection point of the graph of $f$.

c

For what $p$ is the line with equation $y=2x+p$ a tangent to the graph of $f$?

Exercise 6

Somebody wishes to expand his house with a conservatory using four equally sized rectangular frames. The dimensions of each fo these frames are: height $\mathrm{2,5}$ and width $3$. He first studies the possible arrangements where two frames $AB$ and $DE$ are attached perpendicular to the wall. The other two frames $BC$ and $CD$ are placed in such a manner that the area of the floor is maximized.

a

The distance between two frames perpendicular to the wall is $2x$. Show that the area of the floor of the conservatory $A$ corresponds to: $A\left(x\right)=6x+x\cdot \sqrt{9-{x}^{2}}$.

b

Algebraically calculate the maximum floor area for this conservatory.