Question

I will give medal and become a fan. Please help me. It is my last question and I have been staring at it for three hours...
Suppose f (x) and g (x) are differentiable functions with the known values:
derivative with respect to x [g(x)/f(x)] when x = 3
derivative with respect to x [(f(g(x))] when x=4
derivative with respect to x [(x+f(x))^2]
Table of values posted below...

Answer 1

a) Using the quotient rule
f(3)f'(3)-g(3)f'(3)/(f(3))^2 = 5*4-1*2/(5)^2 = 18/25

b) Using the chain rule
f'(g(4) *g(4)= f'(0)*g(4)= 6*6=36

c) derivative with respect to x of the quantity x plus f of x squared when x = 2

Take the derivative of each separately.
[(2+f(x))^2] d/dx(2) + d/dx f(2)
(0)+(1)=1

Answer 2

This is what I think is happening, to answer your first part, but then, I will need confirmation of my input from others before I can continue

Derivative w.r.t \[x\] when \[[g(x)/f(x)]\]

the derivative of a fraction is calculated by the QUOTIENT RULE

\[f(x)/g(x) =\frac{ f'(x)g(x) -g'(x)f(x) }{ g(x)^{2} }\]

Take the figures on the table and fill them in ......BUT AT
\[X=3\]

Answer 3

okay I think that is what I did.