# Various Derivative Problems

• Oct 2nd 2007, 08:11 PM
jwebb19
Various Derivative Problems
I'm having trouble with these problems not sure where to start. Any help is greatly appreciated.

1.) Find all values of https://webwork.math.lsu.edu/webwork...dd0b8b8e91.png where the tangent lines to https://webwork.math.lsu.edu/webwork...7b715f83d1.png and https://webwork.math.lsu.edu/webwork...2d8726df41.png are parallel.

2.) Find an equation for the line tangent to the graph ofhttps://webwork.math.lsu.edu/webwork...a7a84ef741.png at the point https://webwork.math.lsu.edu/webwork...b83c87e881.png for https://webwork.math.lsu.edu/webwork...910c0a5021.png.

3.) Calculate https://webwork.math.lsu.edu/webwork...ae0385ef21.png for https://webwork.math.lsu.edu/webwork...e017108431.png, where https://webwork.math.lsu.edu/webwork...7cc87c1701.png (with a,b,c,d the constants)

Thanks again!
• Oct 2nd 2007, 08:47 PM
Soroban
Hello,jwebb19!

Quote:

1) Find all values of $x$ where the tangent lines to $y = x^4$ and $y = x^5$ are parallel.
We have: . $\begin{array}{ccc}y' & = & 4x^3 \\ y' & = & 5x^4\end{array}$

Since the slopes are equal: . $5x^4 \:=\:4x^3\quad\Rightarrow\quad 5x^4 - 4x^3\:=\:0\quad\Rightarrow\quad x^3(5x - 4)\:=\:0$

Therefore: . $\boxed{x \:=\:0,\:\frac{4}{5}}$

Quote:

2) Find an equation for the line tangent to the graph of $f(x) = -4xe^x$at the point $(1,\,f(1)).$
$f(1)\:=\:-4(1)(e^1) \:=\:-4e$ . . . The point is: . $\left(1,\:-4e\right)$

The slope is: . $f'(x)\;=\;-4xe^x - 4x^x\:=\:-4e^x(x+1)$

When $x = 1$, we have: . $f'(1) \:=\:-4e^1(1 + 1) \:=\;-8e$

The equation is: . $y - (-4e) \:=\:-8e(x - 1) \quad\Rightarrow\quad y + 4e \:=\:-8ex + 8e$

Therefore: . $\boxed{y \;=\;-8ex + 4e}$

Quote:

3) Calculate $y^{(k)}$ .for $0 \leq k \leq 5$

where $y \:=\:9x^4 + ax^3 + bx^2 + cx + d$ (with a,b,c,d constant)

$\begin{array}{ccc}y^{(0)} & = & 9x^4 + ax^3 + bx^2 + cx + d \\
y' & = & 36x^3 + 3ax^2 + 2bx + c \\
y'' & = & 108x^2 + 6x + 2b \\
y''' & = & 216x + 6 \\
y^{(4)} & = & 216 \\
y^{(5)} & = & 0 \end{array}$

• Oct 3rd 2007, 07:06 AM
jwebb19
Thanks a million! I have been trying the following problem to no avail.

If https://webwork.math.lsu.edu/webwork...9a095a6ac1.png then https://webwork.math.lsu.edu/webwork...c032e41041.png

I believe f'(x) = [IMG]file:///C:/Users/Jthan/AppData/Local/Temp/moz-screenshot-8.jpg[/IMG] [sec(x)*tan(x)*(x^3)+3*(x^2)*sec(x)]/[(x^3)^2].

Thanks again!