Find the derivatives of the following functions with respect to corresponding independent variables :
Question 1 :
Differentiate f(x) = x - 3 sinx
Solution :
f(x) = x - 3 sinx
f'(x) = 1 - 3 cos x
Question 2 :
Differentiate y = sin x + cos x
Solution :
f(x) = sin x + cos x
f'(x) = cos x - sin x
Question 3 :
Differentiate f(x) = x sin x
Solution :
f(x) = x sin x
We have to use the product rule to find the derivative.
u = x ==> u' = 1
v = sin x ==> v' = cos x
Product rule :
d(uv) = uv' + vu'
f'(x) = x(cos x) + sin x (1)
f'(x) = x cos x + sin x
Question 4 :
Differentiate y = cos x - 2 tan x
Solution :
f(x) = cos x - 2 tan x
f'(x) = -sin x - 2 sec2 x
Question 5 :
Differentiate g(t) = t3cos t
Solution :
We have to use the product rule to find the derivative.
u = t3 ==> u' = 3t2
v = cos t ==> v' = -sin t
f('x) = t3(-sin t) + cos t (3t2)
f('x) = -t3sin t + 3t2cos t
= t2 (3 cos t - t sin t)
Question 6 :
Differentiate g(t) = 4 sec t + tan t
Solution :
g(t) = 4 sec t + tan t
g'(t) = 4 sec t tan t + sec2 t
Question 7 :
Differentiate y = ex sin x
Solution :
y = ex sin x
u = ex ===> u' = ex
v = sin x ===> v' = cos x
y' = ex (cos x) + sin x(ex)
y' = ex (cos x + sin x)
Question 8 :
Differentiate y = tan x / x
Solution :
y = tan x / x
u = tan x ===> u' = sec2 x
v = x ===> v' = 1
Quotient rule :
d(u/v) = (vu' - uv') / v2
dy/dx = (x sec2 x - tan x (1)) / x2
= (x sec2 x - tan x) / x2
Question 9 :
Differentiate y = sin x / (1 + cos x)
Solution :
y = sin x / (1 + cos x)
u = sin x ===> u' = cos x
v = (1 + cos x) ===> v' = - sin x
Quotient rule :
d(u/v) = (vu' - uv') / v2
dy/dx = ((1 + cos x) cos x - sin x (-sin x)) / (1 + cos x)2
dy/dx = (cos x + cos2 x + sin2 x) / (1 + cos x)2
dy/dx = (1 + cos x) / (1 + cos x)2
dy/dx = 1/(1 + cos x)
Question 10 :
Differentiate y = x / (sin x + cos x)
Solution :
y = x / (sin x + cos x)
u = x ===> u' = 1
v = (sin x + cos x) ===> v' = cos x - sin x
dy/dx = [(sinx+cosx) (1)-x(cosx-sinx)]/(sin x+cosx)2
dy/dx = [sinx + cosx - x cosx + xsinx)]/(sin x+cosx)2
dy/dx = [(1 + x) sinx + (1 - x) cosx]/(sin x+cosx)2
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