Minimizing Time - Need help solving derivative!

[PaintedCavalry]

This is the problem and the example:

Here's where I get to:
d_r= 10-x
d_l= √16+x²

((10-x)/5) + (√16+x²⁾/3)^-1/2 = 0

I just don't know how to solve the derivative. A step by step would be wonderful. I take online classes so I'm pretty much on my own.

 

[HallsofIvy]

If I understand your assignment of variables, what you have is wrong.

If x is the distance (in miles) from the foot of the perpendicular to the point at which he leaves the river (you should say that) then the distance he travels along the river is \(\displaystyle 10- x\) and, at 5 mph, that will require \(\displaystyle \frac{10- x}{5}\) hours. Then he must walk along the hypotenuse of as right triangle with legs of length x and 4 miles and so th e distance he must walk is \(\displaystyle \sqrt{x^2+ 16}\). At mph, that will require \(\displaystyle \frac{\sqrt{x^2+ 16}}{3}= \frac{1}{3}(x^2+ 16)^{1/2}\) hours. So the total time required, which is what you want to minimize, is \(\displaystyle \frac{1}{5}(10- x)+ \frac{1}{3}(x^2+ 16)^{1/2}\), NOT \(\displaystyle \frac{1}{5}(10- x)+ \left(\frac{1}{3}(x^2+ 16)^{1/2}\right)^{-1/2}\) which is what you have.

As for differentiating, you only need to use the chain rule and that the derivative of \(\displaystyle x^n\) is \(\displaystyle nx^{n-1}\).