Test: Differentiability And Chain Rule


10 Questions MCQ Test Mathematics (Maths) Class 12 | Test: Differentiability And Chain Rule


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QUESTION: 1

If f(x) = sin3x, then   = ?

Solution:

QUESTION: 2

The dervative of sin(x2) is

Solution:
QUESTION: 3

If   where V = x2 - 2x + 3 then dy/dx is 

Solution:

y = V3/3
V = x2 - 2x + 3
y = 1/3*v3
dy/dx = d/dx{1/3*(x2-2x+3)3
= 1/3*3*(x2-2x+3)2*(2x-2) 
{d/dx(xn)=nx^n-1*dx/dx} 
= 2(x-1)*(x2-2x+3)2

QUESTION: 4

F(x) = tan (log x)
F'(x) =

Solution:

∫tan(log x) dx
log x = t
x = et
dx = et dt
f(t) = ∫tan t dt
f’(t)= sec2 t
= sec2(log x)

QUESTION: 5

Which of the following function is not differentiable at x = 0?​

Solution:
QUESTION: 6

Solution:
QUESTION: 7

y = log(sec + tan x)

Solution:

y = log(secx + tanx)
dy/dx = 1/(secx + tanx){(secxtanx) + sec2x}
= secx(secx + tanx)/(secx + tanx)
= secx

QUESTION: 8

If y=sin 3x . sin3 x then dy/dx is

Solution:

Given: y = sin 3x . sin3

= 3 sin2x.sin 4x

QUESTION: 9

Solution:

LHD = limh→0 ((−h)2 sin(−1/h) − 0)/−h
limh→0 −h2 sin(1/h)/−h
= limh→0 h × sin(1/h)
= 0
RHD = limh→0 (h2 sin(1/h) − 0)/h
= limh→0 h × sin(1/h)
= 0
RHD=LHD
∴ f(x) is differentiable at x= 0

QUESTION: 10

​Find the derivate of y = sin4x + cos4x

Solution:

y=sin4x,  and z=cos4x
So by using chain rule
df(x)/dx = dsin4x/dx + dcos4x/dx
=dy4/dy * dy/dx + dz4/dz * dzdx
=dy4/dy * dsinxdx + dz4/dz * dcosx/dx
=4y(4−1)⋅cosx+4z(4−1)⋅(−sinx)
=4sin3xcosx − 4cos3xsinx
=4sinxcosx(sin2x − cos2x)
=2sin2x(−cos2x)
=−2sin2xcos2x
=−sin4x

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