"Use the rules for derivatives to find the..." was answered by our community

Stefan Hendricks

Answered question

2022-01-16

Use the rules for derivatives to find the derivative of function defined as follows.

y = 4 x^{2} {(3 x - 2)}^{5}

David Clayton

Beginner2022-01-17Added 36 answers

Step 1
We have to find derivatives of the function:

y = 4 x^{2} {(3 x - 2)}^{5}

We know the formula of derivatives,

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

\frac{d a x^{n}}{d x} = a \frac{{d x}^{n}}{d x}

(where a is constant)

\frac{d x}{d x} = 1

\frac{d a}{d x} = 0

\frac{d (u \cdot v)}{d x} = u \frac{d v}{d x} + v \frac{d u}{d x}

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

Step 2
Differentiating given function with respect to x and applying above formula, we get

\frac{d y}{d x} = \frac{d 4 x^{2} {(3 x - 2)}^{5}}{d x}

= 4 x^{2} \frac{d {(3 x - 2)}^{5}}{d x} + {(3 x - 2)}^{5} \frac{d 4 x^{2}}{d x}

= 4 x^{2} (5 {(3 x - 2)}^{5 - 1}) \frac{d (3 x - 2)}{d x} + {(3 x - 2)}^{5} \times 4 \times \frac{{d x}^{2}}{d x}

= 20 x^{2} {(3 x - 2)}^{4} (3 \frac{d x}{d x} - \frac{d 2}{d x}) + {(3 x - 2)}^{5} \times 4 \times 2 x^{2 - 1}

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Debbie Moore

Beginner2022-01-16Added 43 answers

Step 1

y = 4 x^{2} {(3 x - 2)}^{5}

Step 2

\frac{d y}{d x} = 5 (4 x^{2}) {(3 x - 2)}^{4} (3) + 8 x {(3 x - 2)}^{5}

= {(3 x - 2)}^{4} (84 x^{2} - 16 x)

Result:

\frac{d y}{d x} = {(3 x - 2)}^{4} (84 x^{2} - 16 x)

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