Let `y=ax^(3) + bx^(2) + cx +d`. Find the rate of change of y w.r.t x at x=0.

Answer

To find the rate of change of \(y\) with respect to \(x\) at \(x = 0\) for the function \(y = ax^3 + bx^2 + cx + d\), we will follow these steps:

### Step 1: Write down the function
The given function is:
\[ y = ax^3 + bx^2 + cx + d \]

### Step 2: Differentiate the function with respect to \(x\)
To find the rate of change of \(y\) with respect to \(x\), we need to differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = \frac{d}{dx}(ax^3) + \frac{d}{dx}(bx^2) + \frac{d}{dx}(cx) + \frac{d}{dx}(d) \]
Using the power rule of differentiation, we get:
\[ \frac{dy}{dx} = 3ax^2 + 2bx + c \]

### Step 3: Substitute \(x = 0\) into the derivative
Now, we need to find the value of \(\frac{dy}{dx}\) at \(x = 0\):
\[ \frac{dy}{dx} \bigg|_{x=0} = 3a(0)^2 + 2b(0) + c \]
This simplifies to:
\[ \frac{dy}{dx} \bigg|_{x=0} = 0 + 0 + c = c \]

### Conclusion
Thus, the rate of change of \(y\) with respect to \(x\) at \(x = 0\) is:
\[ \frac{dy}{dx} \bigg|_{x=0} = c \]

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