Differentiate the following functions with respect to x -sin x-x cos x-x sin x-cos x

We have, d/dx(sin x x cos x/x sin x+cos x ) Apply quotient rule, we get =(x sin x+cos x)d/dx(sin x x cos x) (sin x x cos x)d/dx(x sin x+cos x)/(x sin x+cos x)^2 ...

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Byju's Answer

Standard XII

Mathematics

Quotient Rule of Differentiation

Differentiate...

Question

Differentiate the following functions with respect to x :

$\frac{s i n x - x c o s x}{x s i n x + c o s x}$

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Solution

We have,

$\frac{d}{d x} (\frac{s i n x - x c o s x}{x s i n x + c o s x})$

Apply quotient rule, we get

$= \frac{(x s i n x + c o s x) \frac{d}{d x} (s i n x - x c o s x) - (s i n x - x c o s x) \frac{d}{d x} (x s i n x + c o s x)}{(x s i n x + c o s x)^{2}}$

$= \frac{(x s i n x + c o s x) {c o s x - (\frac{d x}{d x} c o s x + c o s x \frac{d x}{d x})} - (s i n x - x c o s x) (\frac{d x}{d x} s i n x + s i n x \frac{d x}{d x}) + \frac{d}{d x} c o s x}{(x s i n x + c o s x)^{2}}$

$= \frac{(x s i n x + c o s x) (c o s x + x s i n x - c o s x) - (s i n x - x c o s x) (x c o s x + s i n x - s i n x)}{(x s i n x + c o s x)^{2}}$

$= \frac{(x s i n x + c o s x) x s i n x - (s i n x - x c o s x) \times c o s x}{(x s i n x + c o s x)^{2}} = \frac{x^{2} s i n^{2} x + x s i n x c o s x - x s i n x c o s x + x^{2} c o s^{2} x}{(x s i n x + c o s x)^{2}}$
$= \frac{x^{2} (s i n^{2} x + c o s^{2} x)}{(x s i n x + c o s x)^{2}} (∵ s i n^{2} x + c o s^{2} = 1)$ $= \frac{x^{2}}{x s i n x + c o s x}$

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Similar questions

Q. Differentiate

\frac{sin x - x cos x}{x sin x + cos x}

Differentiate the following functions with respect to x :

$\frac{x s i n x}{1 + c o s x}$

Find the derivative of the following function with respect to x. $y = \frac{s i n x}{x}$

\int \frac{x^{2} d x}{(x sin x + cos x)^{2}}

is equal to

Question:

if f(x)=x cos x, find f"(x), or d2ydx2

Physics Calculus Differentiation

Solution:

Using the Product Rule, we have f'(x)=xddx(cos x)+cos xddx(x)=-xsinx+cosx
To find f"(x) we differentiate f'(x):
f"(x)=ddx(-x sin x+cos x)=-xddx(sin x)+sin x ddx(-x)+ddx(cos x)
=-x cos x-sin x-sin x = -x cos x-2 sin x

my problem is :

I didn't understand how to solve this question

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Differentiate the following functions with respect to x : sin x−x cos xx sin x+cos x

Differentiate the following functions with respect to x :

$\frac{s i n x - x c o s x}{x s i n x + c o s x}$