Chapter 4: Q10MP (page 239)

If $w = {(r c o s θ)}^{r s i n θ}$ , find $\frac{\partial w}{\partial θ}$ .

Short Answer

Expert verified

The value of $\frac{\partial w}{\partial θ} is [x In x - \frac{y^{2}}{x}] x^{y}$ .

Step by step solution

Given Information

Given that, $w = {(r \cos θ)}^{r \sin θ}$ .

Concept

To find out the value of $\frac{\partial w}{\partial θ}$ take logarithm on both the sides.

Finding the value of ∂w∂θ

Take logarithm on both the sides

$\log w = \log ({(r \cos θ)}^{(r \sin θ)})$

This can also be written as $\log w = (r \sin θ) \log (r \cos θ)$

Now differentiating the above function with respect to

$\frac{1}{w} \frac{\partial w}{\partial θ} = (r \sin θ) \frac{- r \sin θ}{r \cos θ} + \log (r \cos θ) r \cos θ \frac{\partial w}{\partial θ} = [\log (r \cos θ) . r \cos θ - \frac{r^{2} \sin^{2} θ}{r \cos θ}] {(r \cos θ)}^{(r \sin θ)} \frac{\partial w}{\partial θ} = [x In x - \frac{y^{2}}{x}] x^{y}$

Where, $x = r \cos θ, y = r \sin θ$ .

Hence $\frac{\partial w}{\partial θ} = [x In x - \frac{y^{2}}{x}] x^{y}$ .

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If $w = {(r c o s θ)}^{r s i n θ}$ , find $\frac{\partial w}{\partial θ}$ .

Short Answer

Step by step solution

Given Information

Concept

Finding the value of ∂w∂θ

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Ifw=(rcosθ)rsinθ, find∂w∂θ.

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Step by step solution

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If $w = {(r c o s θ)}^{r s i n θ}$ , find $\frac{\partial w}{\partial θ}$ .