DERIVATIVES USING PRODUCT RULE WITH EXAMPLES

About "Derivatives Using Product Rule With Examples"

Derivatives Using Product Rule With Examples :

Here we are going to see how to find the derivatives using product rule.

Let u and v be two differentiable functions. Then

d(uv)/dx  =  u (dv/dx) + v (du/dx)  (or)

d(uv)/dx  =  u v' + v u'

Derivatives using product rule - Examples

Question 1 :

Differentiate y = x sin x cos x

Solution :

Let u  =  x ==>  u'  =  1

v  =  sin x  ==> v'  =  cos x

w  =  cos x  ==>  w'  =  -sin x

d (uvw)  =  u'v w + u v'w + u v w' 

dy/dx  =  1 (sin x)(cos x) + x (cos x) (cos x) + x sinx (-sinx)

  =  sin x cos x + x cos2x - x sin2x

  =  sin x cos x + x (cos2x - sin2x)

  =  sin x cos x + x cos 2x

Question 2 :

Differentiate y = e-x log x

Solution :

Let u  =  e-x ==>  u'  =  -e-x

v  =  log  x  ==> v'  =  1/x

d(uv)  =  u v' + v u'

  =  e-x (1/x) + log x(-e-x)

  =  e-x [(1/x) - log x]

Question 3 :

Differentiate y = (x2 + 5) log (1 + x) e-3x

Solution :

Let u  =  x2 + 5 ==>  u'  =  2x

v  =  log (1 + x)  ==> v'  =  1/(1 + x)

w  =  e-3x  ==>  w'  =  -3 e-3x

d (uvw)  =  u'v w + u v'w + u v w' 

dy/dx 

=  2x(log (1+x))e-3x+(x2+5)(1/(1+x))e-3x+(x2+5)log (1+x) (-3e-3x)

=  e-3x[2x(log (1+x))+(x2+5)/(1+x)-3(x2+5)log (1+x)] 

Question 4 :

Differentiate y = sin x°

Solution :

We know that 1 degree= π/180 radian

so,x degrees= πx/180 radian

Now, the expression is : y = sin( πx/180).

then, dy/dx= π/180.cos( πx/180).

Question 5 :

Differentiate y = log10 x

Solution :

y = log10 x

dy/dx  =  (1/x)log10e  

dy/dx  =  log10e / x

Question 6 :

Draw the function f '(x) if f (x) = 2x2 − 5x + 3

Solution :

f (x) = 2x2 − 5x + 3

f'(x)  =  2(2x) - 5(1) + 0

f'(x)  =  4x - 5

When x = 0, f'(0)  =  -5

When f'(x)  =  0, 4x - 5  =  0  ==>  x  =  5/4  =  1.25  

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