Chapter 4: Q11MP (page 239)

If $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ , find $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ by implicit differentiation.

Short Answer

Expert verified

The first derivative is $\frac{d y}{d x} = \frac{- b^{2}}{a^{2}} (\frac{x}{y})$ and the second derivative is $\frac{d^{2} y}{d x^{2}} = \frac{- b^{4}}{a^{2} y^{3}}$ .

Step by step solution

Given Information

The given equation is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ .

Find the value of dydx

Calculating the first derivative $\frac{d y}{d x}$ then we can use it to arrive at a formula for the second derivative $\frac{d^{2} y}{d x^{2}}$ using the implicit differentiation.

So, finding the first derivative as:

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \frac{2 x}{a^{2}} + \frac{2 y}{b^{2}} (\frac{d y}{d x}) = 0 \frac{d y}{d x} = \frac{- b^{2}}{a^{2}} (\frac{x}{y}) (1)$

Therefore, the first derivative is $\frac{d y}{d x} = \frac{- b^{2}}{a^{2}} (\frac{x}{y})$

Find the value of d2ydx2 .

Now finding $\frac{d^{2} y}{d x^{2}}$ as:

$\frac{d^{2} y}{d x^{2}} = \frac{- b^{2}}{a^{2}} (\frac{y - x \frac{d y}{d x}}{y^{2}})$

Now substituting the value of $\frac{d y}{d x} = \frac{- b^{2}}{a^{2}} (\frac{x}{y})$ as:

$\frac{d^{2} y}{d x^{2}} = \frac{- b^{2}}{a^{2}} (\frac{y - x (\frac{- b^{2} x}{a^{2} y})}{y^{2}}) \frac{d^{2} y}{d x^{2}} = (\frac{\frac{- b^{2}}{a^{2}} y - \frac{- b^{4} x^{5}}{a^{4} y}}{y^{2}}) \frac{d^{2} y}{d x^{2}} = \frac{- b^{2}}{a^{2}} (\frac{y^{2} + \frac{b^{2}}{a^{2}} x^{2}}{y^{3}}) \frac{d^{2} y}{d x^{2}} = \frac{- b^{4}}{a^{2}} (\frac{\frac{y^{2}}{b^{2}} + \frac{x^{2}}{a^{2}}}{y^{3}})$

Now substituting $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ in the above equation as:

$\frac{d^{2} y}{d x^{2}} = \frac{- b^{4}}{a^{2}} (\frac{1}{y^{3}}) \frac{d^{2} y}{d x^{2}} = \frac{- b^{4}}{a^{2} y^{3}}$

Therefore, the first derivative is $\frac{d y}{d x} = \frac{- b^{2}}{a^{2}} (\frac{x}{y})$ and the second derivative is

$\frac{d^{2} y}{d x^{2}} = \frac{- b^{4}}{a^{2} y^{3}}$ .

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If $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ , find $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ by implicit differentiation.

Short Answer

Step by step solution

Given Information

Find the value of dydx

Find the value of d2ydx2 .

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Ifx2a2+y2b2=1 , find dydxandd2ydx2 by implicit differentiation.

Short Answer

Step by step solution

Given Information

Find the value of dydx

Find the value of d2ydx2 .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Engineering Physics

Classical Mechanics

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Study anywhere. Anytime. Across all devices.

If $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ , find $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$ by implicit differentiation.