Q11MP If聽x2a2+y2b2=1 , find dydxand聽... [FREE SOLUTION]

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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91影视

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

Fundamentals of Physics

Classical Mechanics

Fields in Physics

Magnetism

Medical Physics

Further Mechanics and Thermal Physics

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.