Chapter 4: Q4P (page 201)

Given $x = In (u^{2}, v^{2}), u = t^{2}$ and $v = \cos t . find \frac{dx}{dt}$ .

Short Answer

Expert verified

The value of $\frac{d x}{d t}$ is $\frac{4 u t + 2 v \sin t}{u^{2} - v^{2}}$ .

Step by step solution

Explanation of solution

The provided expressions are $x = In (u^{2} - v^{2}), u = t^{2} and v = cost$ .

Chain Rule

The number of functions in the composition affects how many differentiation steps are required when using the chain rule to get the derivative of composite functions.

Calculation

Consider the below equation to find $d x$ .

$d x = \frac{\partial x}{\partial u} d u + \frac{\partial x}{\partial v} d v$

Find $\frac{\partial x}{\partial u}$ .

$\frac{\partial x}{\partial u} = \frac{1}{u^{2} - v^{2}} 2 u = \frac{2 u}{u^{2} - v^{2}}$

Find $\frac{\partial x}{\partial v}$ .

$\frac{\partial x}{\partial v} = \frac{1}{u^{2} - v^{2}} (- 2 v) = \frac{- 2 v}{u^{2} - v^{2}}$

Find dx.

$d x = \frac{- 2 v}{u^{2} - v^{2}} d u - (\frac{2 u}{u^{2} - v^{2}}) d v$

Use the value of $d u = 2 d t$ and $d v = - \sin t d t$ also differentiate above equation with respect to t.

$\frac{d x}{d t} = \frac{2 v}{u^{2} - v^{2}} (2 t) - (\frac{2 v}{u^{2} - v^{2}}) (- \sin t) = \frac{4 u t + 2 v \sin t}{u^{2} - v^{2}}$

Hence the value of $\frac{d x}{d t}$ is $\frac{4 u t + 2 v \sin t}{u^{2} - v^{2}}$ .

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Given $x = In (u^{2}, v^{2}), u = t^{2}$ and $v = \cos t . find \frac{dx}{dt}$ .

Short Answer

Step by step solution

Explanation of solution

Chain Rule

Calculation

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Given x=In(u2,v2),u=t2and v=cost.finddxdt.

Short Answer

Step by step solution

Explanation of solution

Chain Rule

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Energy Physics

Solid State Physics

Oscillations

Wave Optics

Force

Space Physics

Study anywhere. Anytime. Across all devices.

Given $x = In (u^{2}, v^{2}), u = t^{2}$ and $v = \cos t . find \frac{dx}{dt}$ .