Question

Find the first and second derivatives of the function. sqrt of X^2+9

Answer 1

Im assuming you meant:\[\sqrt(x^2 +9)\]
(the nine being inside the square root).
If that is the case, then the you need to use the chain rule, where you will take the derivative of the outside (the square root) and then multiply it by the derivative of whatever is inside (the x^2+9).
I would first rewrite the problem first:
(x^2+9)^(1/2)
since sqrt(y) is the same thing as y^1/2.
Since we need to work outside in let's first consider what the derivative of a sqrt is.
So 1/2(x^2+9)^(1/2-1), secondly, you need to consider the stuff on the inside so the derivative of x^2+9 is 2x. So when we combine these elements, we have
(1/2)(x^2+9)^(-1/2) * 2x
(x^2+9)^(-1/2)*x
x/sqrt(x^2+9)