Differentiation Exercise 1.1

Differentiation Exercise 1.1 12th Class Math Maharashtra Board

Differentiation Exercise 1.1
1. Differentiation Exercise 1.1 12th Class Math Maharashtra Board
2 Differentiate the following w.r.t x

2 Differentiate the following w.r.t x

$(i)cos(x^2+a^2)$

Solution-

$y=cos(x^2+a^2)$

differentiate w.r.t x

$\frac{dy}{dx}=-sin(x^2+a^2)$ $\frac{d}{dx}(x^2+a^2)$

$(ii)\sqrt{e^{3x+2}+5}$

Solution-

differentiate w.r.t x

\frac{d y}{d x} = \frac{1}{\sqrt{e^{3 x + 2} + 5}} \frac{d}{d x} (e^{3 x + 2} + 5)

$\frac{dy}{dx}=\frac{1}{\sqrt{e^{3x+2}+5}}\frac{d}{dx}(e^{3x+2}+5)$

= \frac{1}{2 \sqrt{e^{3 x + 2} + 5}} e^{3 x + 2} \frac{d}{d x} (3 x + 2) + 0

$=\frac{1}{2\sqrt{e^{3x+2}+5}}e^{3x+2}\frac{d}{dx}(3x+2)+0$

= \frac{1}{2 \sqrt{e^{3 x + 2} + 5}} e^{3 x + 2} (3 \times 1 + 0)

$=\frac{1}{2\sqrt{e^{3x+2}+5}}e^{3x+2}(3\times 1+0)$

= \frac{3}{2 \sqrt{e^{3 x + 2} + 5}} e^{3 x + 2}

$=\frac{3}{2\sqrt{e^{3x+2}+5}}e^{3x+2}$

(i i i) \sqrt{t a n \sqrt{t a n (x)}}

$(iii)\sqrt{tan\sqrt{tan(x)}}$

Solution-

differentiate w.r.t x

\frac{d y}{d x} = \frac{1}{2 \sqrt{t a n \sqrt{t a n (x)}}} \times \frac{d}{d x} t a n \sqrt{t a n (x)}

$\frac{dy}{dx}=\frac{1}{2\sqrt{tan\sqrt{tan(x)}}}\times\frac{d}{dx}tan\sqrt{tan(x)}$

= \frac{1}{2 \sqrt{t a n \sqrt{t a n (x)}}} \times s e c^{2} (\sqrt{t a n (x)}) \frac{d}{d x} \sqrt{t a n (x)}

$=\frac{1}{2\sqrt{tan\sqrt{tan(x)}}}\times sec^2(\sqrt{tan(x)})\frac{d}{dx}\sqrt{tan(x)}$

= \frac{1}{2 \sqrt{t a n \sqrt{t a n (x)}}} \times s e c^{2} (\sqrt{t a n (x)}) \frac{1}{2 \sqrt{t a n (x)}} \frac{d}{d x} t a n (x)

$=\frac{1}{2\sqrt{tan\sqrt{tan(x)}}}\times sec^2(\sqrt{tan(x)})\frac{1}{2\sqrt{tan(x)}}\frac{d}{dx}tan(x)$

$\frac{dy}{dx}=\frac{sec^2(\sqrt{tan(x)})}{2\sqrt{tan\sqrt{tan(x)}}}\times \frac{sec^2x}{2\sqrt{tan(x)}}

$(iv)\sqrt{tan\sqrt{x}}$

Solution-

differentiate w.r.t x

$\frac{dy}{dx}=\frac{d}{dx}\sqrt{tan\sqrt{x}}$

$=\frac{1}{2\sqrt{tan\sqrt{x}}}\frac{d}{dx}tan\sqrt{x}$

$=\frac{1}{2\sqrt{tan\sqrt{x}}}sec^2\sqrt{x}\frac{d}{dx}\sqrt x$

$=\frac{1}{2\sqrt{tan\sqrt{x}}}sec^2\sqrt{x}\frac{1}{2\sqrt x}$

$=\frac{sec^2\sqrt{x}}{4\sqrt x\sqrt{tan\sqrt{x}}}$

$(v)cot^3\left[log(x^3)\right]$

differentiate w.r.t x

$\frac{dy}{dx}=\frac{d}{dx}cot^3\left[log(x^3)\right]$

$=3cot^2\left[log(x^3)\right]\frac{d}{dx}\left[log(x^3)\right]$

$=3cot^2\left[log(x^3)\right]\frac{1}{\left[log(x^3)\right]}\frac{d}{dx}(x^3)$

$=3cot^2\left[log(x^3)\right]\frac{3x^2}{\left[log(x^3)\right]}$

$(vi)5^{sin^3x+3}$

Solution-

differentiate w.r.t x

$\frac{dy}{dx}=\frac{d}{dx}(5^{sin^3x+3})$

$=5^{sin^3x+3}\times log 5\times\frac{d}{dx}(sin^3x+3)$

$=5^{sin^3x+3}\times log 5\times(3sin^2x\frac{d}{dx}(sinx)+0)$

$\frac{dy}{dx}=5^{sin^3x+3}\times log 5\times(3sin^2xcosx)$

$(vii)cosec(\sqrt cos x)$

Solution-

differentiate w.r.t x

$\frac{dy}{dx}=\frac{d}{dx}cosec(\sqrt{cos x})$

$=-cosec(\sqrt{cos x})sec(\sqrt{cos x})\frac{d}{dx}(\sqrt{cos x})$

$=-cosec(\sqrt{cos x})sec(\sqrt{cos x})\frac{1}{2\sqrt {cos x}}\frac{d}{dx} cos x$

$=-cosec(\sqrt{cos x})sec(\sqrt{cos x})\frac{-sin x}{2\sqrt {cos x}}$

$=\frac{sin xcosec(\sqrt{cos x})sec(\sqrt{cos x})}{2\sqrt {cos x}}$

$(viii)log[cos(x^3-5)$

Solution-

differentiate w.r.t x

$\frac{dy}{dx}=\frac{d}{dx}(log[cos(x^3-5)])$

$=\frac{1}{cos(x^3-5)}\frac{d}{dx}[cos(x^3-5)]$

$=\frac{1}{cos(x^3-5)}[-sin(x^3-5)\frac{d}{dx}(x^3)-0]$

$=\frac{-3x^2sin(x^3-5)}{cos(x^3-5)}$

$=-3x^2tan(x^3-5)$

$(ix)e^{3 sin^2x-2cos^2x}$

Solution-

differentiate w.r.t x

$\frac{dy}{dx}=\frac{d}{dx}(e^{3 sin^2x-2cos^2x})$

$=e^{3 sin^2x-2cos^2x}\frac{d}{dx}(3 sin^2x-2cos^2x)$

$=e^{3 sin^2x-2cos^2x}\times (6sin x\frac{d}{dx}sinx -4cos x\frac{d}{dx}cos x)$

$=e^{3 sin^2x-2cos^2x}\times(6sinx.cosx+4cosx.sinx)$

$(x)cos^2\left[log(x^2+7)\right]$

Solution-

differentiate w.r.t x

\frac{d y}{d x} = \frac{d}{d x} c o s^{2} [l o g (x^{2} + 7)]

$\frac{dy}{dx}=\frac{d}{dx}cos^2\left[log(x^2+7)\right]$

= 2 c o s [l o g (x^{2} + 7)] \frac{d}{d x} c o s [l o g (x^{2} + 7)]

$=2cos\left[log(x^2+7)\right]\frac{d}{dx}cos\left[log(x^2+7)\right]$

= - 2 c o s [l o g (x^{2} + 7)] s i n [l o g (x^{2} + 7)] \frac{d}{d x} [l o g (x^{2} + 7)]

$=-2cos\left[log(x^2+7)\right]sin\left[log(x^2+7)\right]\frac{d}{dx}\left[log(x^2+7)\right]$

= - 2 c o s [l o g (x^{2} + 7)] s i n [l o g (x^{2} + 7)] \frac{1}{(x^{2} + 7)} \frac{d}{d x} (x^{2} + 7)

$=-2cos\left[log(x^2+7)\right]sin\left[log(x^2+7)\right]\frac{1}{(x^2+7)}\frac{d}{dx}(x^2+7)$

= - 2 c o s [l o g (x^{2} + 7)] s i n [l o g (x^{2} + 7)] \frac{1}{(x^{2} + 7)} (2 x + 0)

$=-2cos\left[log(x^2+7)\right]sin\left[log(x^2+7)\right]\frac{1}{(x^2+7)}(2x+0)$

= - 2 c o s [l o g (x^{2} + 7)] \frac{2 x s i n [l o g (x^{2} + 7)]}{(x^{2} + 7)}

$=-2cos\left[log(x^2+7)\right]\frac{2xsin\left[log(x^2+7)\right]}{(x^2+7)}$

= \frac{- 2 x s i n [l o g (x^{2} + 7)]}{x^{2} + 7}

$=\frac{-2xsin\left[log(x^2+7)\right]}{x^2+7}$

$(xi)tan\left[cos(sin x)\right]$

Solution-

differentiate w.r.t x

\frac{d y}{d x} = \frac{d}{d x} t a n [c o s (s i n x)]

$\frac{dy}{dx}=\frac{d}{dx}tan\left[cos(sin x)\right]$

= s e c^{2} [c o s (s i n x)] \frac{d}{d x} [c o s (s i n x)]

$=sec^2\left[cos(sin x)\right]\frac{d}{dx}\left[cos(sin x)\right]$

= s e c^{2} [c o s (s i n x)] \times - s i n (s i n x) \frac{d}{d x} (s i n x)

$=sec^2\left[cos(sin x)\right]\times-sin(sin x)\frac{d}{dx}(sin x)$

= - s e c^{2} [c o s (s i n x)] \times s i n (s i n x) \times c o s x

$=-sec^2\left[cos(sin x)\right]\times sin(sin x)\times cosx$

$(xii)sec\left[x^4+4\right]$

Solution-

differentiate w.r.t x

\frac{d y}{d x} = \frac{d}{d x} {s e c [t a n (x^{4} + 4)]}

$\frac{dy}{dx}=\frac{d}{dx}\{sec\left[tan(x^4+4)\right]\}$

= s e c [t a n (x^{4} + 4)] t a n [t a n (x^{4} + 4)] \frac{d}{d x} [t a n (x^{4} + 4)]

$=sec\left[tan(x^4+4)\right]tan\left[tan(x^4+4)\right]\frac{d}{dx}\left[tan(x^4+4)\right]$

= s e c [t a n (x^{4} + 4)] t a n [t a n (x^{4} + 4)] s e c^{2} (x^{4} + 4) \frac{d}{d x} (x^{4} + 4)

$=sec\left[tan(x^4+4)\right]tan\left[tan(x^4+4)\right]sec^2(x^4+4)\frac{d}{dx}(x^4+4)$

= s e c [t a n (x^{4} + 4)] t a n [t a n (x^{4} + 4)] s e c^{2} (x^{4} + 4) \times (4 x^{3} + 0)

$=sec\left[tan(x^4+4)\right]tan\left[tan(x^4+4)\right]sec^2(x^4+4)\times(4x^3+0)$

= 4 x^{3} . s e c^{2} (x^{4} + 4) s e c [t a n (x^{4} + 4)] t a n [t a n (x^{4} + 4)]

$=4x^3.sec^2(x^4+4)sec\left[tan(x^4+4)\right]tan\left[tan(x^4+4)\right]$

$(xiii)e^{\left[(log x)^2-log x^2\right]}$

Solution-

differentiate w.r.t x

\frac{d y}{d x} = \frac{d}{d x} e^{[(l o g x)^{2} - l o g x^{2}]}

$\frac{dy}{dx}=\frac{d}{dx}e^{\left[(log x)^2-log x^2\right]}$

= e^{[(l o g x)^{2} - l o g x^{2}]} \frac{d}{d x} [(l o g x)^{2} - l o g x^{2}]

$=e^{\left[(log x)^2-log x^2\right]}\frac{d}{dx}\left[(log x)^2-log x^2\right]$

= e^{[(l o g x)^{2} - l o g x^{2}]} [2 (l o g x) \frac{d}{d x} (l o g x) - \frac{1}{x^{2}} \frac{d}{d x} x^{2}]

$=e^{\left[(log x)^2-log x^2\right]}\left[2(log x)\frac{d}{dx}(log x)-\frac{1}{x^2}\frac{d}{dx}x^2\right]$

= e^{[(l o g x)^{2} - l o g x^{2}]} [2 (l o g x) \frac{1}{x} - \frac{1}{x^{2}} 2 x]

$=e^{\left[(log x)^2-log x^2\right]}\left[2(log x)\frac{1}{x}-\frac{1}{x^2}2x\right]$

= e^{[(l o g x)^{2} - l o g x^{2}]} [\frac{2 (l o g x)}{x} - \frac{2}{x}]

$=e^{\left[(log x)^2-log x^2\right]}\left[\frac{2(log x)}{x}-\frac{2}{x}\right]$

$(xiv)log_{e^2}(log x)$

Solution-

differentiate w.r.t x\\

\frac{dy}{dx}=\frac{d}{dx}sin\sqrt{sin\sqrt{x}}$

= s i n \sqrt{s i n \sqrt{x}} \frac{d}{d x} \sqrt{s i n \sqrt{x}}

$=sin\sqrt{sin\sqrt{x}}\frac{d}{dx}\sqrt{sin\sqrt{x}}$

= s i n \sqrt{s i n \sqrt{x}} \frac{1}{2 \sqrt{s i n \sqrt{x}}} \frac{d}{d x} s i n \sqrt{x}

$=sin\sqrt{sin\sqrt{x}}\frac{1}{2\sqrt{sin\sqrt{x}}}\frac{d}{dx}sin\sqrt{x}$

= \frac{s i n \sqrt{s i n \sqrt{x}}}{2 \sqrt{s i n \sqrt{x}}} c o s \sqrt{x} \frac{d}{d x} \sqrt{x}

$=\frac{sin\sqrt{sin\sqrt{x}}}{2\sqrt{sin\sqrt{x}}}cos\sqrt{x}\frac{d}{dx}\sqrt{x}$

= \frac{s i n \sqrt{s i n \sqrt{x} c o s \sqrt{x}}}{2 \sqrt{s i n \sqrt{x}}} \frac{1}{2 \sqrt{x}}

$=\frac{sin\sqrt{sin\sqrt{x}cos\sqrt{x}}}{2\sqrt{sin\sqrt{x}}}\frac{1}{2\sqrt{x}}$

= \frac{s i n \sqrt{s i n \sqrt{x} c o s \sqrt{x}}}{4 \sqrt{x} \sqrt{s i n \sqrt{x}}}

$=\frac{sin\sqrt{sin\sqrt{x}cos\sqrt{x}}}{4\sqrt{x}\sqrt{sin\sqrt{x}}}$

$(xv)$ log[sec(e^{x}^2)]$

Solution-

differentiate w.r.t x

$(xvi)$ log_{e^2}(log x)$

Solution-

differentiate w.r.t x

\frac{d y}{d x} = \frac{d}{d x} l o g_{e^{2}} (l o g x)

$\frac{dy}{dx}=\frac{d}{dx}log_{e^2}(log x)$

= \frac{l o g (l o g x)}{l o g e^{2}}

$=\frac{log (log x)}{log e^2}$

= \frac{1}{l o g e^{2}} \frac{1}{l o g x} \frac{d}{d x} (l o g x)

$=\frac{1}{log e^2}\frac{1}{log x}\frac{d}{dx}(log x)$

= \frac{1}{l o g x l o g e^{2}} \frac{1}{x}

$=\frac{1}{logx loge^2}\frac{1}{x}$

= \frac{1}{x l o g x l o g e^{2}}

$=\frac{1}{xlogx loge^2}$

= \frac{1}{2 x l o g x}

$=\frac{1}{2xlogx}$

$(xvii)\left[log[log(x)]\right]$

Solution-

differentiate w.r.t x

\frac{d y}{d x} = \frac{d}{d x} {[l o g [l o g (l o g (x))]]}^{2}

$\frac{dy}{dx}=\frac{d}{dx}\left[log\left[log(log(x))\right]\right]^2$

= 2 [l o g [l o g (l o g (x))]] \frac{d}{d x} [l o g [l o g (l o g (x))]]

$=2\left[log\left[log(log(x))\right]\right]\frac{d}{dx}\left[log\left[log(log(x))\right]\right]$

= 2 [l o g [l o g (l o g (x))]] \frac{1}{l o g (l o g (x))} \frac{d}{d x} [l o g (l o g (x))]

$=2\left[log\left[log(log(x))\right]\right]\frac{1}{log(log(x))}\frac{d}{dx}\left[log(log(x))\right]$

= 2 [l o g [l o g (l o g (x))]] \frac{1}{l o g (l o g (x))} \frac{1}{l o g x} \frac{d}{d x} (l o g x)

$=2\left[log\left[log(log(x))\right]\right]\frac{1}{log(log(x))}\frac{1}{log x}\frac{d}{dx}(log x)$

= 2 [l o g [l o g (l o g (x))]] \frac{1}{l o g (l o g (x))} \frac{1}{l o g x} \frac{1}{x}

$=2\left[log\left[log(log(x))\right]\right]\frac{1}{log(log(x))}\frac{1}{log x}\frac{1}{x}$

= \frac{2 [l o g [l o g (l o g (x))]]}{x l o g (l o g x) l o g x}

$=\frac{2\left[log\left[log(log(x))\right]\right]}{xlog(log x)log x}$

$(xviii)sin^2x^2-cos^2x^2$

Solution-

differentiate w.r.t x

\frac{d y}{d x} = \frac{d}{d x} s i n^{2} x^{2} - c o s^{2} x^{2}

$\frac{dy}{dx}=\frac{d}{dx}sin^2x^2-cos^2x^2$

= 2 s i n x^{2} \frac{d}{d x} s i n x^{2} - 2 c o s x^{2} \frac{d}{d x} c o s x^{2}

$=2sinx^2\frac{d}{dx}sinx^2-2cosx^2\frac{d}{dx}cosx^2$

= 2 s i n x^{2} c o s x^{2} \frac{d}{d x} x^{2} + 2 c o s x^{2} s i n x^{2} \frac{d}{d x} x^{2}

$=2sinx^2cosx^2\frac{d}{dx}x^2+2cosx^2sinx^2\frac{d}{dx}x^2$

= 4 x s i n x^{2} c o s x^{2} + 4 x c o s x^{2} s i n x^{2}

$=4xsinx^2cosx^2+4xcosx^2sinx^2$

= 4 x (s i n x^{2} c o s x^{2} + c o s x^{2} s i n x^{2})

$=4x(sinx^2cosx^2+cosx^2sinx^2)$

= 4 x s i n (2 x^{2})

$=4xsin(2x^2)$

Differentiation Exercise 1.1 question 2) pdf

Differentiation Exercise 1.1

Differentiation Exercise 1.1