If y - x sin x -sin x x - find dy - d-

Let y =u +v, where u =x^sin x and v =sin (x^x) ∴dy/dx=du/dx+dv/dx.....(i) Now u =x^sin x⇒ log u =log x^sin x⇒ log u =sin x log x ⇒1/u×du/dx =sin x × ...

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Byju's Answer

Standard XII

Mathematics

Principal Solution of Trigonometric Equation

If y = x sin ...

Question

If $y = x^{s i n x} + s i n (x^{x})$ , find $\frac{d y}{d x}$ .

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Solution

Let $y = u + v, w h e r e u = x^{s i n x} a n d v = s i n (x^{x}) ∴ \frac{d y}{d x} = \frac{d u}{d x} + \frac{d v}{d x} . . . . . (i)$
Now $u = x^{s i n x} \Rightarrow l o g u = l o g x^{s i n x} \Rightarrow l o g u = s i n x l o g x$
$\Rightarrow \frac{1}{u} \times \frac{d u}{d x} = s i n x \times \frac{1}{x} + l o g x c o s x ∴ \frac{d u}{d x} = x^{s i n x} (\frac{s i n x}{x} + l o g x c o s x)$
And, $v = s i n (x^{x}) \Rightarrow \frac{d v}{d x} = c o s (x^{x}) \times \frac{d}{d x} (x^{x}) = c o s (x^{x}) \times \frac{d}{d x} (e^{l o g x^{x}}) = c o s (x^{x}) \times \frac{d}{d x} (e^{x l o g x})$
$\Rightarrow \frac{d v}{d x} = c o s (x^{x}) \times {e^{x l o g x} (x \times \frac{1}{x} + l o g x .1)} ∴ \frac{d v}{d x} = c o s (x^{x}) \times {x^{x} (1 + l o g x)}$
By (i), $\frac{d y}{d x} = x^{s i n x} (\frac{s i n x}{x} + l o g x c o s x) + x^{x} (1 + l o g x) c o s (x^{x}) .$

Suggest Corrections

If y=xsinx+sin(xx), find dydx.

If $y = x^{s i n x} + s i n (x^{x})$ , find $\frac{d y}{d x}$ .