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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