need help with solving this Derivative of products Question

Question

Find an equation for the line tangent to the graph of

f(x)=x3x+5at the point (1,f(1)).

y =

Answer 1

Assuming your function is:

The quotient rules says:

for f (x) = u/v then f '(x) = (u'v - uv')/v²

In this case u = √x which can be written as x^1/2

Using the Power Rule u' = (1/2)x^-1/2 = 1/(2√x)

And v = 3x + 5 so, again by the Power Rule, v' = 3 and v² = (3x + 5)²

Put these all together:

f '(x) = [1/(2√x)•(3x+5) - (√x)(3)]/(3x + 5)²

Simplifying:

f '(x) = [(3x + 5)/(2√x) - 3√x]/(3x + 5)²

f '(x) = [(3x + 5)/(2√x) - 3√x•(2√x)/(2√x)]/(3x + 5)²

f '(x) = [(3x + 5)/(2√x) - 6x/(2√x)]/(3x + 5)²

f '(x) = [(3x + 5 - 6x)/(2√x)]/(3x + 5)²

f '(x) = [(-3x + 5)/(2√x)]/(3x + 5)²

f '(x) = [(-3x + 5)/[2√x)(3x + 5)²]