<s>[INST] Find an equation for the tangent line of $y = e^{4x}$ that is parallel to the linear equation $y = 5 x + 17$. I don't know how to solve this problem:

Find an equation for a tangent line of $y = e^{4x}$ that is parallel to the line given by $y = 5x + 17$.

I have tried with multiple approaches and have gotten 
$$y-e^4(4e^{20})=5(x- 4e^{20}).$$
And my answer remains incorrect any help would be greatly appreciated
 [/INST] The tangent line will have gradient $5$, so you have to find an $x$ such that 
$\frac{d (e^{4x})}{dx} = 5$. Since $\frac{d (e^{4x})}{dx} = 4e^{4x}$, setting
$x = \frac{\log\left(\frac{5}{4}\right)}{4}$ works. Since $\frac{dy}{dx} = 4y$ when $y = e^{4x}$, then $y$ will be $\frac{5}{4}$ when the gradient is $5$.
The line is $y = 5x + c$, where $y$ is $\frac{5}{4}$ when $x$ is $\frac{\log\left(\frac{5}{4}\right)}{4}$. Plugging these values in
gives $\frac{5}{4} = 5\left(\frac{\log\left(\frac{5}{4}\right)}{4}\right) + c$. This means that $c = \frac{5}{4}\left(1 - \log\left(\frac{5}{4}\right)\right)$. So the line is $y = 5x + \frac{5}{4}\left(1 - \log\left(\frac{5}{4}\right)\right)$
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