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'
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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