Question
Differentiate the function using one or more of the differentiation rules. \( y=(3 x+7)^{12} \) \( y^{\prime}=\square \)
Ask by Curry Schultz. in the United States
Feb 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( y' = 36(3x + 7)^{11} \)
Solution
To differentiate the function \( y = (3x + 7)^{12} \), we will use the chain rule. The chain rule states that if you have a composite function \( y = f(g(x)) \), then the derivative \( y' \) is given by:
\[
y' = f'(g(x)) \cdot g'(x)
\]
In this case, let:
- \( f(u) = u^{12} \) where \( u = 3x + 7 \)
- \( g(x) = 3x + 7 \)
Now, we need to find the derivatives of \( f(u) \) and \( g(x) \):
1. Differentiate \( f(u) \):
\[
f'(u) = 12u^{11}
\]
2. Differentiate \( g(x) \):
\[
g'(x) = 3
\]
Now, applying the chain rule:
\[
y' = f'(g(x)) \cdot g'(x) = 12(3x + 7)^{11} \cdot 3
\]
Now, simplifying this expression:
\[
y' = 36(3x + 7)^{11}
\]
Thus, the derivative of the function is:
\[
y' = 36(3x + 7)^{11}
\]
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To differentiate \( y=(3x+7)^{12} \), we can apply the chain rule. According to this rule, if you have a composite function \( f(g(x)) \), the derivative is \( f'(g(x)) \cdot g'(x) \). Here \( f(u) = u^{12} \) and \( g(x) = 3x + 7 \). First, the derivative of \( f(u) \) with respect to \( u \) is \( 12u^{11} \), and substituting back for \( g(x) \), we get \( 12(3x + 7)^{11} \). Next, the derivative of \( g(x) \) is \( 3 \). Now, we multiply these derivatives together: \[ y' = 12(3x + 7)^{11} \cdot 3 = 36(3x + 7)^{11}. \] Thus, \( y' = 36(3x + 7)^{11} \).
