$\displaystyle y=x^2/(1+x) $ Find the derivative of F"(1), By the " it means to find the double derivative, but I have no idea what to do, could anybody help me? thanks
$\displaystyle y=x^2/(1+x) $ Find the derivative of F"(1), By the " it means to find the double derivative, but I have no idea what to do, could anybody help me? thanks
Just in case a picture helps...
... shows the first derivative, where
... is the product rule, straight lines differentiating downwards. But,
... the chain rule, is involved - it hardly leaves a trace in this instance, but we can zoom in on it...
Straight continuous lines still differentiate downwards with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).
Writing the bottom row with a common denominator...
Then a similar routine to differentiate again...
Spoiler:
_________________________________________
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!
your first derivative is incorrect.
$\displaystyle y = \frac{x^2}{1+x}$
quotient rule ...
$\displaystyle y' = \frac{(1+x)(2x) - (x^2)(1)}{(1+x)^2}
$
$\displaystyle y' = \frac{2x + 2x^2 - x^2}{(1+x)^2}$
$\displaystyle y' = \frac{2x+x^2}{(1+x)^2}$
to find the second derivative, quotient rule again ...
$\displaystyle y'' = \frac{(1+x)^2(2+2x) - (2x+x^2) \cdot 2(1+x)}{(1+x)^4}$
no need for simplification from this point ... since you only need to evaluate the second derivative at x = 1, just sub in 1 for x and do the arithmetic.