Find the derivative of y = fractan-1 xsqrtx with respect to x. Find th | Basic Mathematics 2081

Basic Mathematics208110 marksNumerical: derivative and areaDerivatives of inverse trigonometric functions

Find the derivative of $y = \frac{\tan^{-1} x}{\sqrt{x}}$ with respect to $x$ .

Find the area of the region bounded by $y = -x$ and $x = y^2 + 3y$ .

[5+5]

Find the derivative of $y = \frac{\tan^{-1} x}{\sqrt{x}}$ with respect to $x$.

Find the area of the region bounded by $y = -x$ and $x = y^2 + 3y$.

[5+5]

Part 1: $y = \dfrac{\tan^{-1} x}{\sqrt{x}}$ Part 2: Curves $y = -x$ and $x = y^2 + 3y$ --- (a) Derivative Let $u = \tan^{-1} x$ and $v = \sqrt{x} = x^{1/2}$ . $\frac{du}{dx} = \frac{1}{1+x^2}, \qquad \frac{dv}{dx} = \frac{1}{2\sqrt{x}}$ Quotient rule:

Part 1: $y = \dfrac{\tan^{-1} x}{\sqrt{x}}$ Part 2: Curves $y = -x$ and $x = y^2 + 3y$ --- (a) Derivative Let $u = \tan^{-1} x$ and $v = \sqrt{x} = x^{1/2}$. $$\frac{du}{dx} = \frac{1}{1+x^2}, \qquad \frac{dv}{dx} = \frac{1}{2\sqrt{x}}$$ Quotient rule:

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Find the derivative of y = fractan-1 xsqrtx with respect to x. Find the area of the region bounded by y = -x and x = y2 + 3y. [5+5]