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.
1. Answer :
y = (x3 + x2)(x + 5)
Let u = x3 + x2 and v = x + 5.
y = uv
dy/dx = (uv)'
Using product rule of derivative,
dy/dx = uv' + u'v
Substitute u = x3 + x2 and 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)
= x5 + 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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