Please confirm my answer!?

Question

Find the equation of the tangent line to the graph of f at the indicated point. f(X)=1/3X times SQUARE ROOT OF (x^2+5)

Point is (2,2)

Just to make it clear
X is a variable
x^2+5 is in the square root

My answer I recieved it (1/3)x^2/(x^2+5)^(1/2)
I haven't plugged in the points yet. So don't worry about that.

(1/3)x TIMES SQAURE ROOT OF x^2 + 5

x is not at the bottom!

Should it be x^2/3(x^2+5)^1/2, Geezah?

So you use product rule?

THANK YOU!!!

Answer 1

Your derivative is not correct. It appears that you forgot to make use of the product rule. Here is the correct derivation:

d(x√(x²+5)/3)/dx
d(x/3)/dx * √(x²+5) + d(√(x²+5))/dx * x/3
√(x²+5)/3 + 1/(2√(x²+5)) * d(x²)/dx * x/3
√(x²+5)/3 + 2x²/(2√(x²+5)) / 3
√(x²+5)/3 + x²/(3√(x²+5))

Notice the extra term on the left.

Answer 2

EDIT: I misread the function with the division you put in before seeing your correction. So I went back to fix this
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So you're asking just about the derivative, not the final answer?

For the derivative of (x/3) √(x^2 + 5), it should be the derivative of the first term times the second term, plus the first term times the derivative of the second term. I get:

[ (1/3) √(x^2 + 5) ] + [ (x/3) (1/2)(x^2 + 5)^(-1/2) ] =
[ (x^2 + 5) / 3√(x^2 + 5) ] + [ x / 6√(x^2 + 5) ] =
[ 2(x^2 + 5) / 6√(x^2 + 5) ] + [ x / 6√(x^2 + 5) ] =
[ 2(x^2 + 5) + x ] / 6√(x^2 + 5)
[ 2x^2 + x + 10 ] / 6√(x^2 + 5)

Answer 3

the second person's derivative is right. and when i simplified it i got: (2x^2 + 5) / 3*sqrt(x^2+5). now it's easier to plug things in.