Implicit differentiation

Calculus Level 4

x 2 + 25 y 2 = 100 x^2+25y^2=100

Find d 2 y d x 2 \frac{d^2y}{dx^2} of the equation above.

− 4 25 y 3 \dfrac{-4}{25y^3} − 25 y 2 + x 2 625 y 3 \dfrac{-25y^2+x^2}{625y^3} 100 625 y 3 \dfrac{100}{625y^3} − x 25 y \dfrac{-x}{25y}

A Former Brilliant Member by A Former Brilliant Member

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1 solution

x 2 + 25 y 2 = 100 x^2+25y^2=100

By Implicit differentiation, we have

2 x + 50 y d y d x = 0 2x+50y\dfrac{dy}{dx}=0 ⟹ \implies d y d x = − x 25 y \dfrac{dy}{dx}=\dfrac{-x}{25y}

By quotient rule, we have

d 2 y d x 2 = 25 y ( − 1 ) − ( − x ) ( 25 d y d x ) ( 25 y ) 2 \dfrac{d^2y}{dx^2}=\dfrac{25y(-1)-(-x)\left(25\dfrac{dy}{dx}\right)}{(25y)^2}

= − 25 y + 25 x d y d x 625 y 2 =\dfrac{-25y+25x\dfrac{dy}{dx}}{625y^2}

Substitute d y d x = − x 25 y \dfrac{dy}{dx}=\dfrac{-x}{25y}

= − 25 y + 25 x ( − x 25 y ) 625 y 2 =\dfrac{-25y+25x\left(\dfrac{-x}{25y}\right)}{625y^2}

= − 25 y − x 2 y 625 y 2 =\dfrac{-25y-\dfrac{x^2}{y}}{625y^2}

= − 25 y 2 − x 2 y × 1 625 y 2 =\dfrac{-25y^2-x^2}{y} \times \dfrac{1}{625y^2}

= − 25 y 2 − x 2 625 y 3 =\dfrac{-25y^2-x^2}{625y^3}

= − 1 ( 25 y 2 + x 2 ) 625 y 3 =\dfrac{-1(25y^2+x^2)}{625y^3}

substitute x 2 + 25 y 2 = 100 x^2+25y^2=100

= − 100 625 y 3 =\dfrac{-100}{625y^3}

= − 4 25 y 3 =\dfrac{-4}{25y^3}

11 Helpful 0 Interesting 0 Brilliant 0 Confused

You made a typo in the first dy/dx. - 4 should be - x ;)

Peter van der Linden - 3 years, 8 months ago

Same i did but I forget about the about minus sign just before I submit answer. :)

Naren Bhandari - 3 years, 8 months ago

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