Key to "Find the derivatives. y=(x-1)(x+2)(x+3)"

nicekikah

Answered question

2021-11-08

Find the derivatives.
y=(x-1)(x+2)(x+3)

Answer & Explanation

SchepperJ

Skilled2021-11-09Added 96 answers

Step 1
Given function is y=(x−1)(x+2)(x+3).
To find the derivatives of all orders of the function.
Solution:
On simplifying the given function can be written as:
y=(x-1)[x(x+3)+2(x+3)]

= (x - 1) [x^{2} + 3 x + 2 x + 6]

= (x - 1) (x^{2} + 5 x + 6)

= x (x^{2} + 5 x + 6) - 1 (x^{2} + 5 x + 6)

= x^{3} + 5 x^{2} + 6 x - x^{2} - 5 x - 6

= x^{3} + 4 x^{2} + x - 6

So, we have

y = x^{3} + 4 x^{2} + x - 6

.
Step 2
Power rule of differentiation states that:

\frac{d}{d x} (x^{n}) = n x^{n - 1}

Finding first derivative of the function.

y^{'} = \frac{d}{d x} (x^{3} + 4 x^{2} + x - 6)

= 3 x^{2} + 8 x + 1 - 0

= 3 x^{2} + 8 x + 1

Now finding second derivative of the function.

y^{″} = \frac{d}{d x} (3 x^{2} + 8 x + 1)

=3*2x+8*1+0
=6x+8
Step 3
Now finding third derivative of the function.

y^{‴} = \frac{d}{d x} (6 x + 8)

=6*1+0
=6
We know that derivative of a constant function is zero.
So, if we find further derivative then it will be zero.
Hence,

y^{'} = 3 x^{2} + 8 x + 1

, y''=6x+8 and y'''=6.

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