Find out the answer to "Find the derivatives of all orders of the..."

Dillard

Answered question

2021-03-24

Find the derivatives of all orders of the functions

y = (4 x^{2} + 3) (2 - x) x

Demi-Leigh Barrera

Skilled2021-03-26Added 97 answers

Step 1
We first expand the expression

y = (4 x^{2} + 3) (2 - x) x

y = (8 x^{2} - 4 x^{3} + 6 - 3 x) x

y = 8 x^{3} - 4 x^{4} + 6 x - 3 x^{2}

y = - 4 x^{4} + 8 x^{3} - 3 x^{2} + 6 x

Step 2
Then we find the derivatives using the power rule

y = - 4 x^{4} + 8 x^{3} - 3 x^{2} + 6 x

y^{(1)} = - 16 x^{3} + 24 x^{2} - 6 x + 6

y^{(2)} = - 48 x^{2} + 48 x - 6

y^{(3)} = - 96 x + 48

y^{(4)} = - 96

y^{(5)} = 0

All other higher-order derivatives will be 0

alenahelenash

Expert2022-01-24Added 556 answers

Given:

y = (4 x^{2} + 3) (2 - x) x

simplifying above function we get

y = - 4 x^{4} + 8 x^{3} - 3 x^{2} + 6 x

y=f*(x)+kg(x) so sum rule:

\frac{d y}{d x} = \frac{d}{d x} (f (x)) + \frac{d}{d x} (k g (x))

\frac{d (k g (x))}{d x} = k \frac{d (g (x))}{d x}

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

using above rules we get y

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