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

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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 .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physical Quantities And Units

Geometrical and Physical Optics

Thermodynamics

Magnetism

Fundamentals of Physics

Modern Physics

Study anywhere. Anytime. Across all devices.

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

Physical Quantities And Units

Geometrical and Physical Optics

Thermodynamics

Magnetism

Fundamentals of Physics

Modern 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.