Question

What is the equation of the tangent line of `f(x)=sqrt((x+1)^3e^(2x)` at `x=2`?

Answer 1

Numerically, `y = 148.34 - 57.59x`

Explanation:

Alright, so first things first, take the derivative of the function! Remember, the derivative tells us about a rate of change, this will give us the slope! Also, remember, the end goal is the formula `y - y_1 = m(x-x_1)` -- The slope-intercept formula.

So, the derivative. Let's first simplify `f(x)` to
`(x+1)^(3/2) * e^x`

From here, we just have a simple product rule in order to calculate the derivative. Remember, the product rule is: `f'(x)*g(x) + g'(x)*f(x)`

In this case, `(x+1)^(3/2)` is f(x) and `e^x` is g(x).
Doing this, we obtain:
`(3/2*(x+1)^(1/2)*e^x) + (e^x(x+1)^(3/2))`

Now, all we need to do is plug 2 into the derivative in order to calculate the slope!
Doing this, and simplifying, we get:
`(3*sqrt(3)*e^2)/2` + `(e^(2)*3^(3/2))`

It's absolutely disgusting, but it's what we got. For the sake of not wanting to type that out each time from here on out, I'm going to go with it's numerical value, 57.59.

Okay, so we have the slope! Now, let's find the y-coordinate. We can do this by plugging in `x=2` into `f(x)`. Doing this, we have:
`3^(3/2)*e^2` or numerically, 33.16.

Now, all we need to do is plug this into our formula. Doing this we get:
`y - 33.16 = 57.59(2 - x)`

Now, all you need to do is move stuff around! You could do the same with the non-numerical values, it'd just be messier, but more accurate of course. After simplifying, we get:
`y = 148.34 - 57.59x`

And that's our answer!