Answer to Question #283475 in Calculus for Ruchi

Question

Find the second derivative of 3y^4+x^7=5x

Answer 1

3y⁴+x⁷=5x

Differentiating with respect to x

12y³$\frac{dy}{dx}+7x⁶ = 5$

=> 12y³$\frac{dy}{dx}$ = 5 - 7x⁶

=> $\frac{dy}{dx}$ = $\frac{5-7x⁶}{12y³}$ ••••••••(1)

Now 12y³$\frac{dy}{dx}+7x⁶ = 5$

Differentiating again with respect to x

$12y³\frac{d²y}{dx²}+ 36y²{(\frac{dy}{dx})}^2$ + 42x⁵ = 0

=> $2y³\frac{d²y}{dx²}+ 6y²{(\frac{dy}{dx})}^2$ + 7x⁵ = 0

=>

$2y³\frac{d²y}{dx²}+ 6y²{(\frac{5-7x⁶}{12y³})}^2+7x⁵=0$

=> $2y³\frac{d²y}{dx²}= -6y²{(\frac{5-7x⁶}{12y³})}^2-7x⁵$

=> $\frac{d²y}{dx²}= \frac{ -6y²{(\frac{5-7x⁶}{12y³})}^2-7x⁵}{2y³}$

=> $\frac{d²y}{dx²}=- \frac{ 6y²{(\frac{5-7x⁶}{12y³})}^2+7x⁵}{2y³}$

=> $\frac{d²y}{dx²}=- \frac{ 6y²{(\frac{(5-7x⁶)²}{144y⁶})}+7x⁵}{2y³}$

=> $\frac{d²y}{dx²}=- \frac{ {(\frac{(5-7x⁶)²}{24y⁴})}+7x⁵}{2y³}$

=> $\frac{d²y}{dx²}=- \frac{ (5-7x⁶)²+168x⁵y⁴}{48y⁷}$