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Answer :
To find the derivative [tex]\( h^{\prime}(x) \)[/tex] of the function [tex]\( h(x) = 7x^4f(x) \)[/tex], let's use the product rule for differentiation. The product rule states that if you have a function that is the product of two functions, [tex]\( u(x) \)[/tex] and [tex]\( v(x) \)[/tex], then the derivative is:
[tex]\[ \left(u(x) \cdot v(x)\right)^{\prime} = u^{\prime}(x) \cdot v(x) + u(x) \cdot v^{\prime}(x) \][/tex]
In this problem, we have:
- [tex]\( u(x) = 7x^4 \)[/tex]
- [tex]\( v(x) = f(x) \)[/tex]
First, we find the derivatives of these functions:
1. Differentiate [tex]\( u(x) = 7x^4 \)[/tex]:
The derivative of [tex]\( 7x^4 \)[/tex] is:
[tex]\[
u^{\prime}(x) = 28x^3
\][/tex]
2. Differentiate [tex]\( v(x) = f(x) \)[/tex]:
Since [tex]\( f(x) \)[/tex] is an unspecified differentiable function, its derivative is:
[tex]\[
v^{\prime}(x) = f^{\prime}(x)
\][/tex]
Now we apply the product rule:
[tex]\[ h^{\prime}(x) = u^{\prime}(x) \cdot v(x) + u(x) \cdot v^{\prime}(x) \][/tex]
Substituting the derivatives we calculated:
[tex]\[ h^{\prime}(x) = (28x^3) \cdot f(x) + (7x^4) \cdot f^{\prime}(x) \][/tex]
Simplifying gives:
[tex]\[ h^{\prime}(x) = 28x^3f(x) + 7x^4f^{\prime}(x) \][/tex]
Thus, the correct answer is:
A. [tex]\( h^{\prime}(x) = 7x^4f^{\prime}(x) + 28x^3f(x) \)[/tex]
[tex]\[ \left(u(x) \cdot v(x)\right)^{\prime} = u^{\prime}(x) \cdot v(x) + u(x) \cdot v^{\prime}(x) \][/tex]
In this problem, we have:
- [tex]\( u(x) = 7x^4 \)[/tex]
- [tex]\( v(x) = f(x) \)[/tex]
First, we find the derivatives of these functions:
1. Differentiate [tex]\( u(x) = 7x^4 \)[/tex]:
The derivative of [tex]\( 7x^4 \)[/tex] is:
[tex]\[
u^{\prime}(x) = 28x^3
\][/tex]
2. Differentiate [tex]\( v(x) = f(x) \)[/tex]:
Since [tex]\( f(x) \)[/tex] is an unspecified differentiable function, its derivative is:
[tex]\[
v^{\prime}(x) = f^{\prime}(x)
\][/tex]
Now we apply the product rule:
[tex]\[ h^{\prime}(x) = u^{\prime}(x) \cdot v(x) + u(x) \cdot v^{\prime}(x) \][/tex]
Substituting the derivatives we calculated:
[tex]\[ h^{\prime}(x) = (28x^3) \cdot f(x) + (7x^4) \cdot f^{\prime}(x) \][/tex]
Simplifying gives:
[tex]\[ h^{\prime}(x) = 28x^3f(x) + 7x^4f^{\prime}(x) \][/tex]
Thus, the correct answer is:
A. [tex]\( h^{\prime}(x) = 7x^4f^{\prime}(x) + 28x^3f(x) \)[/tex]
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