Questions: Find (d y) for (y=(4 x+5)^2) (d y=) (square) (Simplify your answer.)

Solution Steps

To find \( dy \) for \( y = (4x + 5)^2 \), we need to use the chain rule for differentiation. The chain rule states that if a function \( y \) is composed of an outer function and an inner function, then the derivative of \( y \) with respect to \( x \) is the derivative of the outer function with respect to the inner function, multiplied by the derivative of the inner function with respect to \( x \).

1. Identify the outer function and the inner function.
2. Differentiate the outer function with respect to the inner function.
3. Differentiate the inner function with respect to \( x \).
4. Multiply the results from steps 2 and 3 to get the derivative \( dy \).

We start with the function given by \( y = (4x + 5)^2 \).

To find \( dy \), we differentiate \( y \) with respect to \( x \). Using the chain rule, we find that: \[ \frac{dy}{dx} = 32x + 40 \]

Thus, the differential \( dy \) can be expressed as: \[ dy = (32x + 40) \, dx \]

Final Answer