1. ## 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 where the tangent lines to and are parallel.

2.) Find an equation for the line tangent to the graph of at the point for .

3.) Calculate for , where (with a,b,c,d the constants)

Thanks again!

2. Hello,jwebb19!

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

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

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

3. Thanks a million! I have been trying the following problem to no avail.

If then

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!