Let `y=x^(2/3) + 2x^2`.Find the instantaneous rate of change of y with respect to x at x = 1.

Answer

To find the instantaneous rate of change of \( y \) with respect to \( x \) at \( x = 1 \), we need to follow these steps:

### Step 1: Write down the function
Given:
\[ y = x^{\frac{2}{3}} + 2x^2 \]

### Step 2: Differentiate the function
To find the instantaneous rate of change, we need to compute the derivative \( \frac{dy}{dx} \).

Using the power rule of differentiation:
- The derivative of \( x^{n} \) is \( n \cdot x^{n-1} \).
- For the term \( x^{\frac{2}{3}} \):
\[ \frac{d}{dx}(x^{\frac{2}{3}}) = \frac{2}{3} x^{\frac{2}{3} - 1} = \frac{2}{3} x^{-\frac{1}{3}} \]
- For the term \( 2x^2 \):
\[ \frac{d}{dx}(2x^2) = 2 \cdot 2x^{2-1} = 4x \]

Combining these results, we have:
\[ \frac{dy}{dx} = \frac{2}{3} x^{-\frac{1}{3}} + 4x \]

### Step 3: Evaluate the derivative at \( x = 1 \)
Now, we need to find \( \frac{dy}{dx} \) at \( x = 1 \):
\[ \frac{dy}{dx} \bigg|_{x=1} = \frac{2}{3} \cdot 1^{-\frac{1}{3}} + 4 \cdot 1 \]
Calculating this gives:
\[ \frac{dy}{dx} \bigg|_{x=1} = \frac{2}{3} + 4 \]

### Step 4: Simplify the expression
To combine \( \frac{2}{3} \) and \( 4 \):
Convert \( 4 \) into a fraction with a denominator of \( 3 \):
\[ 4 = \frac{12}{3} \]
Thus,
\[ \frac{dy}{dx} \bigg|_{x=1} = \frac{2}{3} + \frac{12}{3} = \frac{14}{3} \]

### Final Answer
The instantaneous rate of change of \( y \) with respect to \( x \) at \( x = 1 \) is:
\[ \frac{14}{3} \]

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