"Find the derivatives of the functions y=(x+1)2(x2+2x)" was answered by our experts

dictetzqh

Answered question

2021-12-06

Find the derivatives of the functions

y = {(x + 1)}^{2} (x^{2} + 2 x)

Answer & Explanation

Lauren Fuller

Beginner2021-12-07Added 14 answers

Step 1
Given function is

y = {(x + 1)}^{2} (x^{2} + 2 x)

We have to find derivative of given function.
To differentiate the given function we need to use some rules of differentiation.
Product rule of differentiation:

\frac{d}{d x} [f (x) g (x)] = f^{'} (x) g (x) + f (x) g^{'} (x)

Chain rule of differentiation:

\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) g^{'} (x)

Power rule of differentiation:

\frac{d}{d x} [x^{n}] = n x^{n - 1}

Step 2
Differentiating the given function using the above rules,

\frac{d y}{d x} = \frac{d}{d x} [{(x + 1)}^{2} (x^{2} + 2 x)]

= (x^{2} + 2 x) \frac{d}{d x} {(x + 1)}^{2} + {(x + 1)}^{2} \frac{d}{d x} (x^{2} + 2 x)

= (x^{2} + 2 x) \cdot 2 {(x + 1)}^{2 - 1} + {(x + 1)}^{2} (2 x + 2)

= 2 (x^{2} + 2 x) (x + 1) + 2 {(x + 1)}^{2} (x + 1)

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

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

= 2 (x + 1) (2 x^{2} + 4 x + 1)

Hence, required derivative is

\frac{d y}{d x} = 2 (x + 1) (2 x^{2} + 4 x + 1)

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