Math, asked by snaudiyal066, 9 months ago

Differentiate (x2 -- 5x + 8) (x' + 7x + 9) in three ways mentioned belo
(i) by using product rule
(ii) by expanding the product to obtain a single polynomial.
(ii) by logarithmic differentiation.
Do they all give the same answer?​

Answers

Answered by piyushikumari2004
3

Answer:

ANSWER

Let y=(x

2

−5x+8)(x

3

+7x+9)

(i) Let x

2

−5x+8=u and x

3

+7x+9=v

∴y=uv

dx

dy

=

dx

du

.v+u.

dx

dv

By using product rule,

dx

dy

=

dx

d

(x

2

−5x+8).(x

3

+7x+9)+(x

2

−5x+8).

dx

d

(x

3

+7x+9)

dx

dy

=(2x−5)(x

3

+7x+9)+(x

2

−5x+8)(3x

2

+7)

dx

dy

=2x(x

3

+7x+9)−5(x

3

+7x+9)+x

2

(3x

2

+7)−5x(3x

2

+7)+8(3x

2

+7)

dx

dy

=(2x

4

+14x

2

+18x)−5x

3

−35x−45+(3x

4

+7x

2

)−15x

3

−35x+24x

2

+56

dx

dy

=5x

4

−20x

3

+45x

2

−52x+11

(ii) y=(x

2

−5x+8)(x

3

+7x+9)

=x

2

(x

3

+7x+9)−5x(x

3

+7x+9)+8(x

3

+7x+9)

=x

5

+7x

3

+9x

2

−5x

4

−35x

2

−45x+8x

3

+56x+72

=x

5

−5x

4

+15x

3

−26x

2

+11x+72

dx

dy

=

dx

d

(x

5

−5x

4

+15x

3

−26x

2

+11x+72)

=

dx

d

(x

5

)−5

dx

d

(x

4

)+15

dx

d

(x

3

)−26

dx

d

(x

2

)+11

dx

d

(x)+

dx

d

(72)

=5x

4

−5×4x

3

+15×3x

2

−26×2x+11×1+0

=5x

4

−20x

3

+45x

2

−52x+11

(iii) y=(x

2

−5x+8)(x

3

+7x+9)

Taking logarithm on both the sides, we obtain

logy=log(x

2

−5x+8)+log(x

3

+7x+9)

Differentiating both sides with respect to x, we obtain

y

1

dx

dy

=

dx

d

log(x

2

−5x+8)+

dx

d

log(x

3

+7x+9)

y

1

dx

dy

=

x

2

−5x+8

1

.

dx

d

(x

2

−5x+8)+

x

3

+7x+9

1

.

dx

d

(x

3

+7x+9)

dx

dy

=y[

x

2

−5x+8

1

×(2x−5)+

x

3

+7x+9

1

×(3x

2

+7)]

dx

dy

=(x

2

−5x+8)(x

3

+7x+9)[

x

2

−5x+8

2x−5

+

x

3

+7x+9

3x

2

+7

]

dx

dy

=(x

2

−5x+8)(x

3

+7x+9)[

(x

2

−5x+8)(x

3

+7x+9)

(2x−5)(x

3

+7x+9)+(3x

2

+7)(x

2

−5x+8)

]

dx

dy

=2x(x

3

+7x+9)−5(x

3

+7x+9)+3x

2

(x

2

−5x+8)+7(x

2

−5x+8)

dx

dy

=(2x

4

+14x

2

+18x)−5x

3

−35x−45+(3x

4

−15x

3

+24x

2

)+(7x

2

−35x+56)

dx

dy

=5x

4

−20x

3

+45x

2

−52x+11

From the above three observations, it can be concluded that all the

dx

dy

results of

Answered by nsrinivassrinivas14
0

Answer:

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