Question

v) Obtain derivatives of the following
functions:
(i) x sin x
(ii) x4+cos x
(iii) x/sinx
​

Answer 1

Question :

Find the Derivatives of the following:

• x sin x
• x⁴ + cos x
• x / sin x

Formula's Used :

• Genral Formula

$1)\sf\:\frac{d(x {}^{n} )}{dx} = nx {}^{n - 1}$

$2)\sf\:\frac{d(constant)}{dx} = 0$

$3)\sf\dfrac{d(\sin\:x)}{dx}=\cos\:x$

$\sf4)\dfrac{d(\cos\:x)}{dx}=-\sin\:x$

• Quotient rule

Let u = f(x) and v = g(x)

Then ,

$\sf\dfrac{d}{dx}(\dfrac{u}{v})=\dfrac{v\times\frac{du}{dx}-u\times\frac{dv}{dx}}{(v)^2}$

• Chain rule

Let y=f(t) ,t = g(u) and u =m(x) ,then

$\sf\:\dfrac{dy}{dx} = \dfrac{dy}{dt} \times \dfrac{dt}{du} \times \dfrac{du}{dx}$

• Product Rule

Let u = f(x) and v = g(x) , then

$\sf\dfrac{d(uv)}{dx}=u\times\dfrac{dv}{dx}+v\times\dfrac{du}{dx}$

Solution :

We have, to find Derivatives of the following functions :

1) Let y = x sin x

Now Differentiate with respect to x , by product rule

$\sf\dfrac{dy}{dx}=x\times\dfrac{d(\sin\:x)}{dx}+\sin\:x\:\times\dfrac{dx}{dx}$

$\sf\implies\dfrac{dy}{dx}=x\cos\:x+\sin\:x$

2) y = x⁴ + cos x

Now , Differentiate with respect to x , then

$\sf\implies\dfrac{dy}{dx}=\dfrac{d(x^4)}{dx}+\dfrac{d(\cos\:x)}{dx}$

$\sf\implies\dfrac{dy}{dx}=4x^3-\sin\:x$

3) $\sf\:y=\dfrac{x}{\sin\:x}$

Now , Differentiate with respect to x , by quotient rule , then

$\sf\implies\dfrac{dy}{dx}=\dfrac{\sin\:x\:\times\frac{dx}{dx}-x\:\times\frac{d(\sin\:x)}{dx}}{(\sin\:x)^2}$

$\sf\implies\dfrac{dy}{dx}=\dfrac{\sin\:x\:\times1-x\:\times\:cos\:x}{sin^2x}$

$\sf\implies\dfrac{dy}{dx}=\dfrac{\sin\:x-x\:\times\:cos\:x}{sin^2x}$

Answer 2

$\large{\bf{\pink{\underline{Use\:the\:following\:formula\::}}}}$

$1)\:\:\bf{\dfrac{d\:(x^n)}{dx}\:=\:n\:x^{n\:-\:1}\:}$

$2)\:\:\bf{\dfrac{d\:(x)}{dx}\:=\:1\:}$

$3)\:\:\bf{\dfrac{d\:(Constant)}{dx}\:=\:0\:}$

$4)\:\:\bf{\dfrac{d\:(sin\:x)}{dx}\:=\:cos\:x}$

$5)\:\:\bf{\dfrac{d\:(cos\:x)}{dx}\:=\:-\:sin\:x\:}$

Pʀᴏᴅᴜᴄᴛ Rᴜʟᴇ ;-

$\red\bigstar\:\:{\underline{\blue{\boxed{\bf{\purple{\dfrac{d\:\Big(f\:(x)\:g\:(x)\Big)}{dx}\:=\:f\:(x)\:\dfrac{d\Big(g\:(x)\Big)}{dx}\:+\:g\:(x)\:\dfrac{d\Big(f\:(x)\Big)}{dx}\:}}}}}}$

Qᴜᴏᴛɪᴇɴᴛ Rᴜʟᴇ ;-

$\orange\bigstar\:\:{\underline{\green{\boxed{\bf{\green{\dfrac{d}{dx}\:\Big(\dfrac{f\:(x)}{g\:(x)}\Big)\:=\:\dfrac{g\:(x)\:\frac{d\Big(f\:(x)\Big)}{dx}\:-\:f\:(x)\:\frac{d\Big(g\:(x)\Big)}{dx}}{\Big(g\:(x)\Big)^2}\:}}}}}}$

━─━─━─━─━─━─━─━─━─━─━─━─━─━─━

❶ x sinx

Differentiate w.r.t x, we get

$:\implies\:\:\bf{\dfrac{x\:sin\:x}{dx}}$

By using product rule,

$:\implies\:\:\bf{x\times{\dfrac{d\Big(sin\:x\Big)}{dx}}\:+\:sin\:x\times{\dfrac{d\:(x)}{dx}}\:}$

$:\implies\:\:\bf{(x\times{cos\:x})\:+\:(sin\:x\times{1})\:}$

$:\implies\:\:\bf\blue{x\:cos\:x\:+\:sin\:x\:}$

❷ x⁴ + cos x

Differentiate w.r.t x, we get

$:\implies\:\:\bf{\dfrac{x^4\:+\:cos\:x}{dx}}$

$:\implies\:\:\bf{\dfrac{x^4}{dx}\:+\:\dfrac{cos\:x}{dx}}$

$:\implies\:\:\bf{4\:x^{(4\:-\:1)}\:+\:(-\:sin\:x)}$

$:\implies\:\:\bf\orange{4\:x^3\:-\:sin\:x}$

❸ $\bf{\dfrac{x}{sin\:x}}$

Differentiate w.r.t x, we get

$:\implies\:\:\bf{\dfrac{d\:\Big(\frac{x}{sin\:x}\Big)}{dx}}$

By using Quotient rule,

$:\implies\:\:\bf{\dfrac{sin\:x\:\frac{d\:(x)}{dx}\:-\:x\:\frac{d(sin\:x)}{dx}}{(sin\:x)^2}\:}$

$:\implies\:\:\bf{\dfrac{(sinx\times{1})\:-\:(x\times{cos\:x})}{sin^2x}\:}$

$:\implies\:\:\bf{\dfrac{sin\:x\:-\:x\:cos\:x}{sin^2x}\:}$

$:\implies\:\:\bf{\dfrac{sin\:x}{sin^2x}\:-\:\dfrac{x\:cos\:x}{sin^2x}\:}$

$:\implies\:\:\bf{\dfrac{1}{sin\:x}\:-\:\Big(x\times{\dfrac{cos\:x}{sin\:x}}\times{\dfrac{1}{sin\:x}}\Big)\:}$

$:\implies\:\:\bf{cosec\:x\:-\:x\:.\:{cot\:x}\:.\:{cosec\:x}\:}$

$:\implies\:\:\bf\green{cosec\:x\:\Big(1\:-\:x\:{cot\:x}\Big)\:}$