Math Assignment Class XII Ch-5

Math Assignment Class XII Chapter 5 Derivatives

Question 1 Use chain rule to find the derivative of

$y=\left(\frac{2x-1}{2x+1}\right)^{2}\;\;\;\:Ans:\frac{8(2x-1)}{(2x+1)^{3}}$

Question 2. Differentiate the following w.r.t. x

$y=log_{10}x+log_{x}10+log_{x}x+log_{10}10$

Answer

$\frac{dy}{dx}=\frac{1}{xlog10}-\frac{log10}{x(logx)^{2}}$

Solution Hint

$\frac{logx}{log10}+\frac{log10}{logx}+\frac{logx}{logx}+\frac{log10}{log10}$

$=\frac{logx}{log10}+\frac{log10}{logx}+1+1$

Now differentiating w. r. t. x and taking log10 as constant.

Question 3. Differentiate the following w. r. t. x at x = 1

$y=e^{x(1+logx)}$

Answer: 2e

Question 4.

$If\:\:\frac{x}{x-y}=log\frac{a}{x-y}$ , Then prove that : $\frac{dy}{dx}=\frac{2y-x}{y}$

Question 5: Differentiate log(x e^x) w. r. t. xlogx.

Answer 5: $\frac{1+x}{x(1+logx)}$

Question 6: Differentiate x² w. r. t. x³. ........... Ans: 2/3x.

Question 7: $If\:e^{x}+e^{y}=e^{x+y}$ , Prove that $\:\:\frac{dy}{dx}+e^{y-x}=0$

Solution Hint:

$e^{x}+e^{y}=e^{x+y}$

Dividing on both side by $e^{x+y}$

$\frac{e^{x}}{e^{x+y}}+\frac{e^{y}}{e^{x+y}}=1$

$\Rightarrow e^{-y}+e^{-x}=1$

Now differentiating w. r. t. x we get the required answer.

Question 8: Differentiate (x^x)^x w. r. t. x

Answer

$\frac{dy}{dx}=x^{x^{2}}[x+2xlogx]$

Solution Hint:

$(x^{x})^{x}=x^{x^{2}}$ Now differentiating w. r. t. x

Question9. Differentiate x¹⁶y⁹= (x²+ y)¹⁷ w. r. t. x ,

Answer : $\frac{dy}{dx}=\frac{2y}{x}$

Question10) Differentiate (logx)^x + x^logx, w. r. t. x.

Answer

$(logx)^{x}\left[\frac{1}{logx}+log(logx)\right]+2x^{logx-1}.logx$

Question11

$If\:\:f(x)=|cosx|,find\:f'\left(\frac{3\pi}{4}\right)$

Answer: $\frac{1}{\sqrt{2}}$

Question 12:

$If\:\:y=\sqrt{cosx+\sqrt{cosx+\sqrt{cos+....\infty}}}$

Then prove that : $(2y-1)\frac{dy}{dx}=-sinx$

Solution 12:

$y=\sqrt{cosx+\sqrt{cosx+\sqrt{cos+....\infty}}}$

$y=\sqrt{cosx+y}\:\:\Rightarrow\:\:y^{2}=cosx+y$

Differentiating both side w. r. t. x we get

$2y\frac{dy}{dx}=-sinx+\frac{dy}{dx}$

$(2y-1)\frac{dy}{dx}=-sinx$

Question 13. Find dy/dx at x = 1, y = π/4, If Sin²y + cosxy = k

Answer 13. $\frac{\pi(\sqrt{2}+1)}{4}$

Question 14. If x = a(2θ - sin2θ) and y = a(1-cos2θ), find dy/dx, when θ = π/4.
Answer 14: [ 1]
Question 15. Differentiate Log(cosex) w. r. t x

Ans[ -cotx ].
Question 16.

$If\:y=\sqrt{tanx+\sqrt{tanx+\sqrt{tanx+......\infty}}}$ then prove that

$(2y-1)\frac{dy}{dx}=Sec^{2}x$
Question 17.

$If\:y=\frac{sin^{-1}x}{\sqrt{1-x^{2}}},$ then prove that $(1-x^{2})\frac{d^{2}y}{dx^{2}}-3x\frac{dy}{dx}-y=0$

Question 18.

$If\:x=cost+log\:tan\frac{t}{2},\:y=sint$ , then find the value of $\frac{d^{2}y}{dx^{2}}\:at\:\:t=\frac{\pi}{4}$

$Ans:\: 2\sqrt{2}$
Question 19: (a) Differentiate the following w. r. t. x

$y=sin^{-1}\left(\frac{2^{x+1}}{1+4^{x}}\right)$

Ans: $\frac{2^{x+1}}{1+4^{x}}log\:2$

Question 19 (b)

If $\mathbf{y=log\left[tan\left(\frac{\pi}{4}+\frac{x}{2}\right)\right]$ , then prove that $\mathbf{\frac{dy}{dx}-secx=0$

Question 20. Differentiate the following w. r. t. x

$y=sin^{-1}\left\{\frac{2^{x+1}.3^{x}}{1+(36)^{x}}\right\}$ Ans: $\frac{2(log6)6^{x}}{1+(36)^{x}}$

Question 21. Differentiate: $y=e^{e^{x}}\:w.\:r.\:t.\:x.$

Ans: $e^{e^{x}}\times e^{x}$

Question 22.

Find the value of k if the following function is continuous.

$\mathbf{f(x)=\left\{\begin{matrix}\frac{kcosx}{\pi-2x}\;\;,\;\;x<\frac{\pi}{2}\\\\\frac{3tan2x}{2x-\pi}\;\;,\;\;x>\frac{\pi}{2}\end{matrix}\right.}$

Answer: k = 6

Question 23.

Find the value of k for which the following function is continuous at x = 0

$\mathbf{f(x)=\left\{\begin{matrix}\frac{\sqrt{1+kx}-\sqrt{1-kx}}{x}\;\;,\;\;-1\leq x<0\\\\\frac{2x+1}{x-2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:,\:\:\:\:\:\:0\leq x<1\end{matrix}\right.}$

Answer: k = -1/2
Question 24:

Show that f(x) = |x - 2| is continuous but not differentiable at x = 2.

Question 25:

Discuss the continuity and differentiability of the following function.
$\mathbf{f(x)=\left\{\begin{matrix}1-x\:\:\:\:\:\:\:\:,\:\:x<1\\(1-x)(2-x),\:1\leq x\leq 2\\3-x\:\:\:\:\:,\:\:\:\:\:x>2\end{matrix}\right.}$

Answer:

f(x) is continuous at x = 1 but not continuous at x = 2
f(x) is differentiable at x = 1.
As f(x) is not continuous at x = 2 so f(x) is not differentiable at x = 2.

Question 26

Check the differentiability of the function

$f(x)=\left\{\begin{matrix}x,&x<1\\2-x&1\leq x\leq 2&\;\;at\;\;\;x=2\\-2+3x-x^{2},&x>2\\\end{matrix}\right.$

Solution Hint

$LHD=\displaystyle\lim_{x\to a^{-}}\frac{f(x)-f(a)}{x-a}=-1$

$RHD=\displaystyle\lim_{x\to a^{+}}\frac{f(x)-f(a)}{x-a}=-1$

LHD = RHD ⇒ f(x) is differentiable at x = 2

Question 27 (a)

Question Check the differentiability of

$\mathbf{f(x)=\left\{\begin{matrix}x^{2}+1,&0\leq x<1\\3-x,&1\leq x\leq 2\\\end{matrix}\right.\;\;at\;x=1}$

Solution Hint:

At x = 1, LHD = 2 and RHD = -1

As LHD ≠ RHD, this function is not differentiable

Question 27 (b)

If a function defined by $\mathbf{f(x)=\left\{\begin{matrix}kx+1,&x\leq\pi\\cosx,&x>\pi\\\end{matrix}\right.$ is continuous at x = π, then find the value of k

Answer: -2/π

Question 28

If f(x) = (x + 1)^cotx is continuous at x = 0 then find f(0).

Answer = e

Solution: We know that

$\displaystyle\lim_{x\to 0}f(x)=f(0)$

Now, f(x) = (x+1)^cotx

Log[f(x)] = Log[(x+1)^cotx ]

Log[f(x)] = cotx Log(x+1)

$\displaystyle\lim_{x\to 0}Log[f(x)]=\displaystyle\lim_{x\to 0}cotx\;log(x+1)$

$Log\displaystyle\lim_{x\to 0}f(x)=\displaystyle\lim_{x\to 0}\left(\frac{x}{tanx}\right)\;\left(\frac{log(x+1)}{x}\right)$

$Log\displaystyle\lim_{x\to 0}f(x)=1\times 1=1$
$\displaystyle\lim_{x\to 0}f(x)=e^{1}$
$\displaystyle\lim_{x\to 0}f(x)=e=f(0)$

Question 29 : Find $\frac{d}{dx}|x|$

Solution: $\frac{d}{dx}|x|=\frac{d}{dx}\sqrt{x^{2}}$

$\frac{d}{dx}|x|=\frac{1}{2\sqrt{x^{2}}}\times 2x=\frac{x}{|x|}$

Question 30 :

Check whether the function f(x) = x² |x| is differentiable at x = 0 or not.

Solution Hint

This function can be written as $f(x)=\left\{\begin{matrix}x^{3},&x\geq 0\\-x^{3},&x\leq 0\\\end{matrix}\right.$

At x = 0, Find LHD and RHD we get

LHD = RHD = 0

Yest this function is differentiable.

Question 31 :

If $\mathbf{y=\sqrt{tan\sqrt{x}}}$ , then find dy/dx

Answer: $\mathbf{\frac{dy}{dx}}=\frac{1+y^{4}}{4y\sqrt{x}}$

Question 33 :

If $\mathbf{\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y)}$ then prove that $\mathbf{\frac{dy}{dx}=\sqrt{\frac{1-b^{2}}{1-x^{2}}}}$

Solution Hint for the Q. No. 33 to Q. No. 40

Question 34 :

If y = cosec(cot^-1x), then prove that

$\mathbf{\sqrt{1+x^{2}}\frac{dy}{dx}-x=0}$

Question 35 :

If x = e^cos3t and y = e^sin3t , prove that $\mathbf{\frac{dy}{dx}=\frac{-ylogx}{xlogy}}$

Question 36 :

If x = e^x/y, then prove that $\mathbf{\frac{dy}{dx}=\frac{logc-1}{(logx)^{2}}}$

Question 37:

If y = cos³(sec²2t), find dy/dt

Question 38

If x^y = e^{x - y}, prove that $\mathbf{\frac{dy}{dx}=\frac{logx}{(1+logx)^{2}}}$

Question 39

Given that y = (sinx)^x . x^sinx + a^x, find dy/dx

Question 40

If x = a sin³θ, y = bcos³θ, then find $\mathbf{\frac{d^{2}y}{dx^{2}}}$ at θ = π/4

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