Question

If g(x) = 2f(x) + x + log (f(x)) then $\frac{d\left(g\right(x\left)\right)}{d\left(f\right(x\left)\right)}$_________

• A. $2+\frac{1}{f\left(x\right)}$
• B. $2+\frac{1}{f\left(x\right)}+\frac{1}{f^{\prime }\left(x\right)}$
• C. $2+\frac{1}{f\left(x\right)f^{\prime }\left(x\right)}+\frac{1}{f^{\prime }\left(x\right)}$
• D. None of the above

Answer 1

The correct option is B $2+\frac{1}{f\left(x\right)}+\frac{1}{f^{\prime }\left(x\right)}$

We discussed in the videos that if $y=f\left(x\right)andz=\varphi \left(x\right)$
then $\frac{dy}{dx}=\frac{f^{\prime }\left(x\right)}{{\varphi }^{\prime }\left(X\right)}$
Same thing we will apply here
$\frac{d\left(g\right(x\left)\right)}{d\left(f\right(x\left)\right)}=\frac{g^{\prime }\left(x\right)}{f^{\prime }\left(x\right)}g\left(x\right)=2f\left(x\right)+x+log\left(f\right(x\left)\right)g^{\prime }\left(x\right)=2f^{\prime }\left(x\right)+1+\frac{1}{f\left(x\right)}x{f}^{\prime }\left(x\right)$
$So\frac{g^{\prime }\left(x\right)}{f^{\prime }\left(x\right)}willbe\frac{2f^{\prime }\left(x\right)+1+\frac{f^{\prime }\left(x\right)}{f\left(x\right)}}{f^{\prime }\left(x\right)}=2+\frac{1}{f^{\prime }\left(x\right)}+\frac{1}{f\left(x\right)}$ (Note that Chain rule is applied here for log(f(x))
which is the correct answer.