Differentiate the following questions w-r-t- x- e x-sin x

Let y = e^x/sin x. Differenatiate both sides w.r.t. x, we get dy/dx=d/dx(e^x/sin x ) [ Using Quotient rule d/dx ( u/v )=vd/dxu ud/dxv/v^2] =sin xd/dxe^x ...

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Byju's Answer

Standard XII

Mathematics

Domain and Range of Basic Inverse Trigonometric Functions

Differentiate...

Question

Differentiate the following questions w.r.t. x.

$\frac{e^{x}}{s i n x}$

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Solution

Let y = $\frac{e^{x}}{s i n x}$ . Differenatiate both sides w.r.t. x, we get

$\frac{d y}{d x} = \frac{d}{d x} (\frac{e^{x}}{s i n x}) [U s i n g Q u o t i e n t r u l e \frac{d}{d x} (\frac{u}{v}) = \frac{v \frac{d}{d x} u - u \frac{d}{d x} v}{v^{2}}]$

$= \frac{s i n x \frac{d}{d x} e^{x} - e^{x} \frac{d}{d x} s i n x}{s i n^{2} x} = \frac{s i n x e^{x} - e^{x} c o s x}{s i n^{2} x} = \frac{e^{x} (s i n x - c o s x)}{s i n^{2} x}, x \neq n π, n ϵ Z .$

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Differentiate the following questions w.r.t. x. exsin x

Let y = exsin x. Differenatiate both sides w.r.t. x, we get dydx=ddx(exsin x) [Using Quotient rule ddx(uv)=vddxu−uddxvv2] =sin xddxex−exddxsin xsin2x=sin x ex−ex cos xsin2x=ex(sin x−cos x)sin2x,x≠nπ,nϵZ.

Differentiate the following questions w.r.t. x.

$\frac{e^{x}}{s i n x}$