Question

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d) \\(y = 4x^4(3x^2 - 1)\\)

\\(u = 4x^4\\) \t \\(v = 3x^2 - 1\\)
\\(u = 16x^3\\) \t \\(v = 6x\\)

\\(y = 6x(4x^4) + 16x^3(3x^2 - 1)\\)

Explanation:

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<reasoning>

Identify the function and components

The given function is:
\[ y = 4x^4(3x^2 - 1) \]
We can define the two factors as:
\[ u = 4x^4 \]
\[ v = 3x^2 - 1 \]

Find the individual derivatives

Using the power rule for differentiation:
\[ u' = \frac{d}{dx}(4x^4) = 16x^3 \]
\[ v' = \frac{d}{dx}(3x^2 - 1) = 6x \]

Apply the product rule formula

The product rule states:
\[ y' = u v' + v u' \]
Substituting our functions and their derivatives:
\[ y' = (4x^4)(6x) + (3x^2 - 1)(16x^3) \]
This matches the expression written in the image:
\[ y' = 6x(4x^4) + 16x^3(3x^2 - 1) \]

Simplify the derivative expression

Now we expand and combine like terms to find the fully simplified derivative:
\[ y' = 24x^5 + 16x^3(3x^2) - 16x^3 \]
\[ y' = 24x^5 + 48x^5 - 16x^3 \]
\[ y' = 72x^5 - 16x^3 \]

Factor the final simplified result

We can factor out the greatest common factor, which is \(8x^3\):
\[ y' = 8x^3(9x^2 - 2) \]
</reasoning>

<answer>
\[ y' = 72x^5 - 16x^3 = 8x^3(9x^2 - 2) \]
</answer>

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Answer:

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<reasoning>

Identify the function and components

The given function is:
\[ y = 4x^4(3x^2 - 1) \]
We can define the two factors as:
\[ u = 4x^4 \]
\[ v = 3x^2 - 1 \]

Find the individual derivatives

Using the power rule for differentiation:
\[ u' = \frac{d}{dx}(4x^4) = 16x^3 \]
\[ v' = \frac{d}{dx}(3x^2 - 1) = 6x \]

Apply the product rule formula

The product rule states:
\[ y' = u v' + v u' \]
Substituting our functions and their derivatives:
\[ y' = (4x^4)(6x) + (3x^2 - 1)(16x^3) \]
This matches the expression written in the image:
\[ y' = 6x(4x^4) + 16x^3(3x^2 - 1) \]

Simplify the derivative expression

Now we expand and combine like terms to find the fully simplified derivative:
\[ y' = 24x^5 + 16x^3(3x^2) - 16x^3 \]
\[ y' = 24x^5 + 48x^5 - 16x^3 \]
\[ y' = 72x^5 - 16x^3 \]

Factor the final simplified result

We can factor out the greatest common factor, which is \(8x^3\):
\[ y' = 8x^3(9x^2 - 2) \]
</reasoning>

<answer>
\[ y' = 72x^5 - 16x^3 = 8x^3(9x^2 - 2) \]
</answer>

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