Question Find the derivative of $y=2 x^{2}(3-2 x)$ using the product rule.

Studdy Solution

STEP 1

1. The function $y=2x^2(3-2x)$ is differentiable, and we can apply the product rule to find its derivative.
2. The product rule states that if $u(x)$ and $v(x)$ are differentiable functions, then the derivative of their product $u(x)v(x)$ is given by $u'(x)v(x) + u(x)v'(x)$ .

STEP 2

1. Identify the two functions that are being multiplied (the product).
2. Differentiate each function separately.
3. Apply the product rule to find the derivative of the entire expression.
4. Simplify the resulting expression.

STEP 3

Identify the two functions that are being multiplied. In this case, $u(x) = 2x^2$ and $v(x) = (3-2x)$ .

STEP 4

Differentiate $u(x) = 2x^2$ with respect to $x$ .
$u'(x) = \frac{d}{dx}(2x^2) = 2 \cdot 2x^{2-1} = 4x$

STEP 5

Differentiate $v(x) = (3-2x)$ with respect to $x$ .
$v'(x) = \frac{d}{dx}(3-2x) = 0 - 2 = -2$

STEP 6

Apply the product rule to find the derivative of $y = u(x)v(x)$ . According to the product rule, the derivative $y'$ is given by:
$y' = u'(x)v(x) + u(x)v'(x)$
Substitute $u(x)$ , $v(x)$ , $u'(x)$ , and $v'(x)$ into this formula.

STEP 7

Substitute the differentiated functions into the product rule formula:
$y' = (4x)(3-2x) + (2x^2)(-2)$

STEP 8

Simplify the resulting expression by distributing and combining like terms.
$y' = 12x - 8x^2 - 4x^2$

STEP 9

Combine the like terms to get the final simplified derivative.
$y' = 12x - 12x^2$
The derivative of $y=2x^2(3-2x)$ with respect to $x$ is $y' = 12x - 12x^2$ .

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