PRODUCT RULE OF DERIVATIVE WORKSHEET

Question 1 :

Find dy/dx, if y = (x3 + x2)(x + 5).

Question 2 :

Find dy/dx, if y = (-2x4 - 3)(-2x2 + 1).

Question 3 :

Find dy/dx, if y = (x3/3)(x3 - 5).

Question 4 :

Find dy/dx, if y = √x(x2 - 2x + 3).

Question 5 :

Find dy/dx, if y = exlnx.

Question 6 :

Find dy/dx, if y = 2xx5.

Question 7 :

Find dy/dx, if y = 2xlnx.

Question 8 :

Find dy/dx, if y = x3sinx.

Question 9 :

Find dy/dx, if y = e2xtanx.

Question 10 :

Find dy/dx, if y = 3sin3xcosx.

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Answers

1. Answer :

y = (x3 + x2)(x + 5)

Let u = x3 + xand v = x + 5.

y = uv

dy/dx = (uv)'

Using product rule of derivative,

dy/dx = uv' + u'v

Substitute u = x3 + xand v = x + 5.

dy/dx = (x3 + x2)(x + 5)' + y + (x3 + x2)'(x + 5)

= (x3 + x2)(1 + 0) + (3x3 - 1 + 2x2 - 1)(x + 5)

= (x3 + x2)(1) + (3x2 + 2x)(x + 5)

= x3 + x2 + (3x3 + 15x2 + 2x2 + 10x)

= x3 + x2 + (3x3 + 17x2 + 10x)

= x3 + x2 + 3x3 + 17x2 + 10x

= 4x3 + 18x2 + 10x

2. Answer :

y = (-2x4 - 3)(-2x2 + 1)

Let u = -2x4 - 3 and v = -2x2 + 1.

y = uv

dy/dx = (uv)'

Using product rule of derivative,

dy/dx = uv' + u'v

Substitute u = -2x4 - 3 and v = -2x2 + 1.

dy/dx = (-2x4 - 3)(-2x2 + 1)' + (-2x4 - 3)'(-2x2 + 1)

= (-2x4 - 3)(-2 ⋅ 2x2 - 1 + 0) + (-2 ⋅ 4x4 - 1 - 0)(-2x2 + 1)

= (-2x4 - 3)(-4x) + (-8x3)(-2x2 + 1)

= (-2x4 - 3)(-4x) - 8x3(-2x2 + 1)

= 8x5 + 12x + 16x5 - 8x3

= 24x5 - 8x3 + 12x

3. Answer :

y = (x3/3)(x3 - 5)

Let u = x3/3 and v = x3 - 5.

y = uv

dy/dx = (uv)'

Using product rule of derivative,

dy/dx = uv' + u'v

Substitute u = x3/3 and v = x3 - 5.

dy/dx = (x3/3)(x3 - 5)' + (x3/3)'(x2 - 5)

= (x3/3)(3x3 - 1 - 0) + (3x3 - 1/3)(x2 - 5)

= (x3/3)(3x2) + (3x2/3)(x2 - 5)

= x3x2 + x2(x2 - 5)

= x+ x4- 5x2

4. Answer :

y = √x(x2 - 2x + 3)

Let u = √x and v = x2 - 2x + 3.

y = uv

dy/dx = (uv)'

Using product rule of derivative,

dy/dx = uv' + u'v

Substitute u = √x and v = x2 - 2x + 3.

dy/dx = √x(x2 - 2x + 3)' + (√x)'(x2 - 2x + 3)

= √x(x2 - 2x + 3)' + (x1/2)'(x2 - 2x + 3)

5. Answer :

y = exlnx

Let u = ex and v = lnx.

y = uv

dy/dx = (uv)'

Using product rule of derivative,

dy/dx = uv' + u'v

Substitute u = ex and v = lnx.

dy/dx = ex(lnx)' + (ex)'lnx

6. Answer :

y = 2xx5

Let u = 2x and v = x5.

y = uv

dy/dx = (uv)'

Using product rule of derivative,

dy/dx = uv' + u'v

Substitute u = 2x and v = x5.

dy/dx = 2x(x5)' + (2x)'x5

= 2x(5x5 - 1) + (2xln2)x5

= 2x5x4 + x52xln2

= x42x(5 + xln2)

7. Answer :

y = 2xlnx

Let u = 2x and v = lnx.

y = uv

dy/dx = (uv)'

Using product rule of derivative,

dy/dx = uv' + u'v

Substitute u = 2x and v = lnx.

dy/dx = 2x(lnx)' + (2x)'lnx

8. Answer :

y = x3sinx

Let u = x3 and v = sinx.

y = uv

dy/dx = (uv)'

Using product rule of derivative,

dy/dx = uv' + u'v

Substitute u = x3 and v = sinx.

dy/dx = x3(sinx)' + (x3)'sinx

= x3(cosx) + (3x3 - 1)sinx

= x3cosx + 3x2sinx

= x2(xcosx + 3sinx)

9. Answer :

y = e2xtanx

Let u = e2x and v = tanx.

y = uv

dy/dx = (uv)'

Using product rule of derivative,

dy/dx = uv' + u'v

Substitute u = e2x and v = tanx.

dy/dx = e2x(tanx)' + (e2x)'tanx

= e2x(sec2x) + (2e2x)tanx

= e2x(sec2x + 2tanx)

10. Answer :

y = 3sin3xcosx

Let u = sin3x and v = cosx.

y = 3uv

dy/dx = 3(uv)'

Using product rule of derivative,

dy/dx = 3(uv' + u'v)

Substitute u = sin3x and v = cosx.

dy/dx = 3[sinx3x(cosx)' + (sin3x)'cosx]

= 3[sin3x(-sinx) + (3sin3-1xcosx)cosx]

= 3[-sin3xsinx + (3sin2xcosx)cosx]

= 3[-sin3xsinx + 3sin2xcos2x]

= -3sin3xsinx + 9sin2xcos2x

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