if y=[a^2+b] sin[x^2] where a and b are constant then the value f dy/dx is

To find the derivative of the function \( y = [a^2 + b] \sin(x^2) \) with respect to \( x \), we can apply the product rule of differentiation. Since \( a \) and \( b \) are constants, they will remain unchanged during the differentiation process. Let's break this down step by step.

Understanding the Components

The function can be viewed as a product of two parts: a constant term \( [a^2 + b] \) and the function \( \sin(x^2) \). The product rule states that if you have two functions \( u \) and \( v \), then the derivative of their product \( uv \) is given by:

• \( \frac{d(uv)}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \)

In our case, we can let:

• \( u = a^2 + b \) (a constant)
• \( v = \sin(x^2) \)

Applying the Product Rule

Since \( u \) is a constant, its derivative \( \frac{du}{dx} \) is zero. Therefore, we only need to differentiate \( v \). The derivative of \( \sin(x^2) \) can be found using the chain rule:

• \( \frac{dv}{dx} = \cos(x^2) \cdot \frac{d(x^2)}{dx} \)
• \( \frac{d(x^2)}{dx} = 2x \)

So, we have:

• \( \frac{dv}{dx} = \cos(x^2) \cdot 2x \)

Combining the Results

Now, substituting back into the product rule formula, we get:

• \( \frac{dy}{dx} = (a^2 + b) \cdot \frac{dv}{dx} + \sin(x^2) \cdot 0 \)
• \( \frac{dy}{dx} = (a^2 + b) \cdot (2x \cos(x^2)) \)

Final Expression

Thus, the derivative \( \frac{dy}{dx} \) simplifies to:

\( \frac{dy}{dx} = 2x(a^2 + b) \cos(x^2) \)

This expression gives you the rate of change of \( y \) with respect to \( x \). It combines the constant factors with the derivative of the sine function, showcasing how the output of the function varies as \( x \) changes.