Question
The instantaneous rate of change of \( f(x)=\sqrt{6 x+7} \) at \( x=3 \) is \( \ldots \ldots \ldots \ldots \). \( \frac{3}{5} \) \( \frac{3}{5} \)
Ask by Mccoy Wright. in Egypt
Mar 08,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The instantaneous rate of change of \( f(x) = \sqrt{6x+7} \) at \( x=3 \) is \( \frac{3}{5} \).
Solution
1. Write the function as
\[
f(x) = (6x+7)^{\frac{1}{2}}.
\]
2. Differentiate using the chain rule:
\[
f'(x) = \frac{1}{2}(6x+7)^{-\frac{1}{2}} \cdot 6 = \frac{3}{\sqrt{6x+7}}.
\]
3. Evaluate the derivative at \( x = 3 \):
\[
f'(3) = \frac{3}{\sqrt{6 \cdot 3 + 7}} = \frac{3}{\sqrt{18+7}} = \frac{3}{\sqrt{25}} = \frac{3}{5}.
\]
Thus, the instantaneous rate of change of the function at \( x=3 \) is \(\frac{3}{5}\).
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Simplify this solution Extra Insights
To find the instantaneous rate of change of \( f(x) = \sqrt{6x + 7} \) at \( x = 3 \), we need to calculate the derivative \( f'(x) \). The derivative is given by \( f'(x) = \frac{3}{5} \) at that point, so the rate of change is \( \frac{3}{5} \). But did you know that derivatives are a cornerstone of calculus, helping us understand how functions behave? They tell us not just the slope of a curve at a specific point but also can help optimize real-world situations like maximizing profits or minimizing costs. It’s like having a secret recipe to decode any changing situation!
