Questions: Find the derivative of the function. Do not find the product before finding the derivative. y=5 x^2(4 x^4-3 x^2) Choose the correct answer below. A. y'=(4 x^4-3 x^2)(16 x^3-6 x)+(10 x)(5 x^2) B. y'=(5 x^2)(4 x^4-3 x^2)+(16 x^3-6 x)(10 x) C. y'=(5 x^2)(16 x^3-6 x)+(4 x^4-3 x^2)(10 x) D. y'=(4 x)(5 x^2)+(4 x^4-3 x^2)(16 x^3-6 x)

Find the derivative of the function. Do not find the product before finding the derivative.
y=5 x^2(4 x^4-3 x^2)

Choose the correct answer below.
A. y'=(4 x^4-3 x^2)(16 x^3-6 x)+(10 x)(5 x^2)
B. y'=(5 x^2)(4 x^4-3 x^2)+(16 x^3-6 x)(10 x)
C. y'=(5 x^2)(16 x^3-6 x)+(4 x^4-3 x^2)(10 x)
D. y'=(4 x)(5 x^2)+(4 x^4-3 x^2)(16 x^3-6 x)

Transcript text: Find the derivative of the function. Do not find the product before finding the derivative. \[ y=5 x^{2}\left(4 x^{4}-3 x^{2}\right) \] Choose the correct answer below. A. $y^{\prime}=\left(4 x^{4}-3 x^{2}\right)\left(16 x^{3}-6 x\right)+(10 x)\left(5 x^{2}\right)$ B. $y^{\prime}=\left(5 x^{2}\right)\left(4 x^{4}-3 x^{2}\right)+\left(16 x^{3}-6 x\right)(10 x)$ C. $y^{\prime}=\left(5 x^{2}\right)\left(16 x^{3}-6 x\right)+\left(4 x^{4}-3 x^{2}\right)(10 x)$ D. $y^{\prime}=(4 x)\left(5 x^{2}\right)+\left(4 x^{4}-3 x^{2}\right)\left(16 x^{3}-6 x\right)$

Solution

Solution Steps

To find the derivative of the function $ y = 5x^2(4x^4 - 3x^2) $, we will use the product rule. The product rule states that if you have a function $ y = u(x)v(x) $, then the derivative $ y' = u'(x)v(x) + u(x)v'(x) $. Here, let $ u(x) = 5x^2 $ and $ v(x) = 4x^4 - 3x^2 $. We will find the derivatives $ u'(x) $ and $ v'(x) $, and then apply the product rule.

Step 1: Define the Functions

We start with the function given in the problem: \[ y = 5x^2(4x^4 - 3x^2) \] We can identify the two parts of the product: \[ u = 5x^2 \quad \text{and} \quad v = 4x^4 - 3x^2 \]

Step 2: Calculate the Derivatives

Next, we compute the derivatives of $ u $ and $ v $: \[ u' = \frac{d}{dx}(5x^2) = 10x \] \[ v' = \frac{d}{dx}(4x^4 - 3x^2) = 16x^3 - 6x \]

Step 3: Apply the Product Rule

Using the product rule, which states that $ y' = u'v + uv' $, we substitute our derivatives and original functions: \[ y' = (10x)(4x^4 - 3x^2) + (5x^2)(16x^3 - 6x) \]

Step 4: Simplify the Expression

Now we can simplify the expression for $ y' $: \[ y' = 10x(4x^4 - 3x^2) + 5x^2(16x^3 - 6x) \] This can be expanded as: \[ y' = 40x^5 - 30x^3 + 80x^5 - 30x^3 \] Combining like terms gives: \[ y' = (40x^5 + 80x^5) - (30x^3 + 30x^3) = 120x^5 - 60x^3 \]