Question
The Derivative (R) Question \( 2,9,4.11 \) Part 1 of 3 Find the indicated quantities for \( f(x)=3 x^{2} \).
Ask by Wilson Murray. in the United States
Jan 27,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The derivative of \( f(x) = 3x^{2} \) is \( f'(x) = 6x \).
Solution
Sure, let's work through **Part 1** of your derivative question for the function \( f(x) = 3x^{2} \).
### **1. Find the Derivative \( f'(x) \)**
To find the derivative of \( f(x) = 3x^{2} \), we'll use the **power rule** of differentiation. The power rule states:
\[
\frac{d}{dx} [x^n] = n \cdot x^{n-1}
\]
Let's apply this step-by-step:
1. **Identify the components:**
- **Coefficient (constant factor):** 3
- **Variable part:** \( x^2 \)
- **Exponent (n):** 2
2. **Apply the power rule:**
\[
f'(x) = \frac{d}{dx} [3x^{2}] = 3 \cdot \frac{d}{dx} [x^{2}]
\]
\[
\frac{d}{dx} [x^{2}] = 2x^{2-1} = 2x
\]
3. **Combine the results:**
\[
f'(x) = 3 \cdot 2x = 6x
\]
### **Summary**
- **Original Function:** \( f(x) = 3x^{2} \)
- **First Derivative:** \( f'(x) = 6x \)
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If **Part 1** includes additional "indicated quantities," feel free to provide more details, and I'll be happy to assist further!
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Simplify this solution The Deep Dive
The derivative of the function \( f(x) = 3x^2 \) is found by applying the power rule. By decreasing the exponent by one and multiplying by the original exponent, we have \( f'(x) = 2 \cdot 3x^{2-1} = 6x \). So, the rate of change of the function at any point \( x \) can be determined using \( f'(x) = 6x \). To explore the behavior of this function, you can plug in various values of \( x \) into \( f'(x) \). For instance, at \( x = 1 \), \( f'(1) = 6(1) = 6 \), indicating the function’s steepness at that point. Additionally, at \( x = 0 \), \( f'(0) = 6(0) = 0 \), showing that the function has a horizontal tangent, a key concept in understanding the shape of the curve!
