So this is a pretty straightforward derivative problem. The definition of a derivative is the slope of the tangent line to the curve, so yeah.

We first find the derivative of the function.


There might be some cool trick to do this, but Quotient Rule should suffice.




We could simplify this by factoring out an , resulting in . However, this isn't necessary.

This is the general derivative for the function, but now we have to evaluate it at the value to find the slope of the tangent line there. The point given is .

Evaluating:

\[ \dfrac{d}{dx} \left[\dfrac{e^x}{x}\right] \right|_{x=1} \, = \dfrac{e^{(1)} (1) -e^{(1)}}{(1)^2} \]

Yay we love exponents of .



But remember we're not done yet. This is only the slope of the tangent line. The problem asks for the equation of the tangent line.

So what do we have? We are given the point, and by calculating the derivative, we find the slope.

point+slope =......Point-Slope form!
By the way

We plug our given values in. Point: Slope: 0





gg