Chain Rule Examples With Solutions :
Here we are going to see how we use chain rule in differentiation.
Question 1 :
Differentiate f(x) = x/√(7 - 3x)
Solution :
u = x
u' = 1
v = √(7 - 3x)
v' = 1/2√(7 - 3x)(-3) ==> -3/2√(7 - 3x)==>-3/2√(7 - 3x)
f'(x) = [√(7 - 3x)(1) - x(-3/2√(7 - 3x))]/(√(7 - 3x))2
f'(x) = [√(7 - 3x) + (3x/2√(7 - 3x))]/(√(7 - 3x))2
f'(x) = [2(7 - 3x) + 3x)/2√(7 - 3x))]/(7 - 3x)
f'(x) = (14-6x+3x)/(2(7-3x)√(7-3x))
f'(x) = (14-3x)/[2(7-3x)√(7-3x)]
Question 2 :
Differentiate y = tan(cos x)
Solution :
y = tan(cos x)
Let u = cos x
Differentiate the function "u" with respect to "x"
du/dx = -sin x
y = tan u
Differentiate the function "y" with respect to "x".
dy/dx = sec2 u (du/dx)
dy/dx = sec2 (cos x)(-sin x)
dy/dx = -sin x sec2 (cos x)
Question 3 :
Differentiate y = sin2x/cos x
Solution :
u = sin2x ==> u' = 2 sin x cos x
v = cos x ==> v' = - sinx
dy/dx = (cos x(2 sin x cos x) - sin2x (- sinx)) / (cos2x)
dy/dx = (2 sin x cos2 x + sin3x) / (cos2x)
dy/dx = 2 sin x + (sin3x / cos2x)
= 2 sin x + tan2x sin x
= sin x (2 + tan2x)
dy/dx = sin x (1 + sec2x)
Question 4 :
Differentiate y = 5-1/x
Solution :
Let u = -1/x
du/dx = -1/x2
y = 5u
dy/dx = 5u (log 5) (du/dx)
= 5-1/x (log 5) ( -1/x2)
dy/dx = (-5-1/x log 5)/x2
Question 4 :
Differentiate y = √(1 + 2 tan x)
Solution :
Let u = (1 + 2 tan x)
du/dx = 0 + 2 sec2x ==> 2 sec2x
y = √u
dy/dx = (1/2√u) (du/dx)
dy/dx = (1/2√(1 + 2 tan x) )(2 sec2x)
dy/dx = (sec2x/√(1 + 2 tan x))
Question 5 :
Differentiate y = sin3x + cos3x
Solution :
dy/dx = 3 sin2x(cos x) + 3 cos2x(-sin x)
dy/dx = 3 sinx cos x (sin x - cos x)
Question 6 :
Differentiate y = sin2 (cos kx)
Solution :
Let u = cos kx
Differentiate the function "u" with respect to "x"
du/dx = -sin kx (k) ==> - k sin kx
y = sin2 u
Differentiate the function "y" with respect to "x"
dy/dx = 2 sin u cos u (du/dx)
= sin 2u (du/dx)
= sin (2 cos kx) (-k sin kx)
dy/dx = -k sin kx sin (2 cos kx)
After having gone through the stuff given above, we hope that the students would have understood, "Chain Rule Examples With Solutions"
Apart from the stuff given in "Chain Rule Examples With Solutions", if you need any other stuff in math, please use our google custom search here.
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
©All rights reserved. onlinemath4all.com
Apr 23, 24 09:10 PM
Apr 23, 24 12:32 PM
Apr 23, 24 12:07 PM