$\text { Evaluate } \frac { d y } { d x } \text { for the equation } x ^ { \frac { 4 } { 5 } } + y ^ { \frac { 4 } { 5 } } = 257 \text { at the given point } ( 1024,1 ) \text {. Round }$ your answer to two decimal places.
A) $\frac { d y } { d x } = 16.00$
B) $\frac { d y } { d x } = 4.00$
C) $\frac { d y } { d x } = - 0.25$
D) $\frac { d y } { d x } = 0.25$
E) $\frac { d y } { d x } = - 0.06$

Question

Quizplus · Accepted Answer

To find $\frac{dy}{dx}$, we differentiate both sides of the equation $x^{\frac{4}{5}} + y^{\frac{4}{5}} = 257$ with respect to $x$, using implicit differentiation. This gives us $\frac{4}{5}x^{-\frac{1}{5}} + \frac{4}{5}y^{-\frac{1}{5}}\frac{dy}{dx} = 0$. Solving for $\frac{dy}{dx}$ at the point $(1024,1)$, we find $\frac{dy}{dx} = -\frac{x^{-\frac{1}{5}}}{y^{-\frac{1}{5}}} = -\frac{1024^{-\frac{1}{5}}}{1^{-\frac{1}{5}}} = -\frac{1}{4} = -0.25$.

Your Answer to Two Decimal Places dydx=16.00\frac { d y } { d x } = 16.00dxdy​=16.00

Your Answer to Two Decimal Places $\frac { d y } { d x } = 16.00$