Question

i need help:
find the derivative dy/dx.
y= x^(1/2)*(e^(x^2))*(x^2+1)^10

Answer 1

is it
\[
y =\sqrt x e^{x^2} ( x^2+1)^{10}
\]

Answer 2

If it is use, the product rule for derivative
\[g= u v w\\
g'= u' vw + v' uw + w' u v

\]

Answer 3

yes it is

Answer 4

So apply the product rule above and you will be done.

Answer 5

but how to do with (x^2+1)^10

Answer 6

20 x ( x^2 +1)^9

Answer 7

Your final answer should look like
\[
\frac{e^{x^2}
\left(x^2+1\right)^{10}}{2
\sqrt{x}}+2 e^{x^2} x^{3/2}
\left(x^2+1\right)^{10}+20
e^{x^2} x^{3/2}
\left(x^2+1\right)^9
\]

Answer 8

thanks i see

Answer 9

An alternative in case you encounter some more of these, plus quotients (probably):
Take the ln of the function you're differentiating and use properties of ln to simplify:
\[y =\sqrt x e^{x^2} ( x^2+1)^{10}\\
\ln{y}=\frac{1}{2}\ln{x}+x^2+10\ln(x^2+1)\]
Differentiate wrt x and multiply by y:
\[\frac{y'}{y}=\frac{1}{2x}+2x+\frac{20x}{x^2+1}\\
y'=\sqrt x e^{x^2} ( x^2+1)^{10}\left( \frac{1}{2x}+2x+\frac{20x}{x^2+1}\right)\]