Q1.If $ f(2)=4,f{}'(2)=4$ then find the value of $ lim_{x\rightarrow 2}\frac{xf(2)-2f(x)}{x-2}$. [AIEEE 2002]
a. 2
b.-4
c.2
d.3
Ans: -4
Hint: Let, $\small x-2=h$, Now find $\small \lim_{h\rightarrow 0} \frac{(h+2)f(20-2f(h+2)}{h}$.
Q2.If $ f(1)=1,f{}'(1)=2$ then find the value of $\small lim_{x\rightarrow 1}\frac{\sqrt f(x) -1}{\sqrt x -1}$ [AIEEE 2002]
a. 1
b. 2
c. 3
d. 4
Ans: 2
Hint: $f{}'(1)= lim_{x\rightarrow 1} \frac{f(x)-f(1)}{x-1}$, Find $lim_{x\rightarrow 1}\frac{f(x) -1}{x -1}\frac{\sqrt x +1}{\sqrt f(x) +1}$= $ 2 \times \frac{2}{2}$
Q3.If $y=tan^{-1}\sqrt\frac{1-sinx}{1+sinx}$ then find the value of $\frac{dy}{dx}$ at $ x=\frac{\pi}{6}$[WBJEE 2004]
a. 1
b.-1
c. $\frac{1}{2}$
d. -2
Ans: $\frac{1}{2}$
Hint: $\frac {1-sinx}{1+sinx}= tan^{2}(\frac{\beta}{2}), x+\beta=\frac{\pi}{2}=$
Q4.If $y=tan^{-1}\frac{cosx}{1+sinx}$ then find the value of $\frac{dy}{dx}$.
a. 1
b.-1
c. $ \frac{1}{2}$
d. -2
Ans: $\frac{1}{2}$
Hint: Apply the formula $\small \frac{cos2x}{1+sin2x}=\frac{1-tanx}{1+tanx}=\frac{tan 45^o-tanx}{1+tan 45^otanx}$
Q5.$ x^y=e^{x-y}$.$\frac{dy}{dx}=?$ [WBJEE 1983]
Ans:
Hint:
$y logx =x-y$
or, $ y\frac{1}{x}+logx \frac{dy}{dx}=1-\frac{dy}{dx}$
Q6. $x=e^{y+e^{y+e^{y+e^{y+...\infty}}}}$.$ \frac{dy}{dx}=?$ [AIEEE 2003]
Ans:
Hint:
$ x=e^{y+x}$
or, $ logx={y+x}$
Q7. $ f(x)=(\frac{a+x}{b+x})^x+cos x$.then find the $f{(0)}'=?$
Ans: $\small f(x)=e^{x log(\frac{a+x}{b+x})}+cos x$
Apply chain rule
Q8.$f(x)=(1+x)(1+x^2)(1+x^4)(1+x^8)$. Then $ f{(1)}'=?$
Ans:
Hint:
$f(x)=\frac{(1-x)(1+x)(1+x^2)(1+x^4)(1+x^8)}{1-x}$
or,$ f(x)=\frac{(1-x^2)(1+x^2)(1+x^4)(1+x^8)}{1-x}$
or,$f(x)=\frac{(1-x^{16})}{1-x}$
or,$ f(x)=1+2x+3x^2+.....+15x^4$
Q9. $ y=log_{sinx}secx+10^{x^2}$. Find $ \frac{dy}{dx}=?$[WBJEE 1983]
Ans:
Hint:
$\small log_{sinx}secx=\frac{log(secx)}{log(sinx)}=log(secx)-log(sinx)$.Apply Chain rule.
Q10.$f(x)=log_{5}log_{3}x$ then find $\small f{(e)}'=?$
a. $ \frac{1}{elog5}$
b. $ \frac{5}{elog5}$
c. $\frac{1}{elog3}$
d. $ \frac{1}{log5}$
Ans: $ \frac{1}{elog5}$
Hint: $ log5 log_{3}x=log5\frac{logx}{log3} = log5(logx-log3)$.
Find $ f{(x)}'$ at the point x=e
Q11. If $g,f$ are inverse function $ f{(x)}'=\frac{1}{1+x^3}$. Then find the value of $ g{(x)}'=?$.
Ans:
Hint:
$ f^{-}(x)=g(x)$
or,$x=f[g(x)]$
or,$ \frac{dx}{dx}=f^{'}[g(x)]g^{'}(x)$
or,$ 1=f^{'}[g(x)]g^{'}(x)$
So, $g^{'}(x)=1+[g(x)]^3$
Q11. If $g,f$ are inverse function $ f{(x)}'=\frac{1}{1+x^3}$. Then find the value of $ g{(x)}'=?$.
Ans:
Hint:
$ f^{-}(x)=g(x)$
or,$x=f[g(x)]$
or,$ \frac{dx}{dx}=f^{'}[g(x)]g^{'}(x)$
or,$ 1=f^{'}[g(x)]g^{'}(x)$
So, $g^{'}(x)=1+[g(x)]^3$
Q12.If $y=log(log(logx)),\frac{dy}{dx}=?$
a.$log(logx)$
b.$log(logx)$
c.$ \frac{1}{xlogxlog(logx)}$
d.0
Ans: c
Hint: Chain rule
Q13.If $f(x+y)=f(x)f(y)$ for $x,y$ belongs to $R$, $ f(5)=2,f^{'}(0)=3$.Find $f^{'}(5)=?$
a.5
b.3
c.2
d.6
Ans: d
Hint: If$ x=0,y=0$, then $ f(0)={f(0)}^{2}$. So, $ f(0)=1$
$f^{'}(0)=lim_{h \rightarrow 0}\frac{f(0+h)-f(0)}{h}=3$
$f^{'}(5)$
=$ lim_{h \rightarrow 0}\frac{f(5+h)-f(5)}{h}$
=$im_{h \rightarrow 0}\frac{f(5)f(h)-f(5)}{h}$
=6
Q14. $y=f[f(x)],f(0)=0,f^{'}(0)=3$ then $ {dy}{dx}=?$
a.3
b.6
c.9.
d.0
Ans: 9
Hint: Chain rule
Q13.If $f(x+y)=f(x)f(y)$ for $x,y$ belongs to $R$, $ f(5)=2,f^{'}(0)=3$.Find $f^{'}(5)=?$
a.5
b.3
c.2
d.6
Ans: d
Hint: If$ x=0,y=0$, then $ f(0)={f(0)}^{2}$. So, $ f(0)=1$
$f^{'}(0)=lim_{h \rightarrow 0}\frac{f(0+h)-f(0)}{h}=3$
$f^{'}(5)$
=$ lim_{h \rightarrow 0}\frac{f(5+h)-f(5)}{h}$
=$im_{h \rightarrow 0}\frac{f(5)f(h)-f(5)}{h}$
=6
Q14. $y=f[f(x)],f(0)=0,f^{'}(0)=3$ then $ {dy}{dx}=?$
a.3
b.6
c.9.
d.0
Ans: 9
Hint: Chain rule
Q15.If $f(x)=x^3-x^2+x$ and find $\frac{d}{dx}{f^{-}(x)}=?$
at $ x=f(2)$.
a. $ \frac{1}{3}$
b.$\frac{1}{9}$
c.$\frac{1}{27}$
d.$0$
Ans: $ \frac{1}{9}$
Hint: Let $g(x)=f^{-}(x)$
So, $g[f(x)]=x$
or, $g^{'}[f(x)]f^{'}(x)=1$
Put $x=2$
or,$g^{'}[f(2)]f^{'}(2)=1$
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