d) Find \( \frac{d y}{d x} \) if \( y=\frac{\sin x^{\cos x}}{x^{3}} \)

To find \( \frac{dy}{dx} \) for the function \( y = \frac{\sin(x^{\cos x})}{x^3} \), we will use the quotient rule and the chain rule. The quotient rule states that if \( y = \frac{u}{v} \), then \[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \] In our case, let: - \( u = \sin(x^{\cos x}) \) - \( v = x^3 \) First, we need to find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \). 1. **Finding \( \frac{dv}{dx} \)**: \[ \frac{dv}{dx} = \frac{d}{dx}(x^3) = 3x^2 \] 2. **Finding \( \frac{du}{dx} \)**: To differentiate \( u = \sin(x^{\cos x}) \), we will use the chain rule. Let \( z = x^{\cos x} \), then \( u = \sin(z) \). Using the chain rule: \[ \frac{du}{dx} = \cos(z) \frac{dz}{dx} \] Now we need to find \( \frac{dz}{dx} \): \[ z = x^{\cos x} = e^{\cos x \ln x} \] Using the chain rule and product rule: \[ \frac{dz}{dx} = e^{\cos x \ln x} \cdot \frac{d}{dx}(\cos x \ln x) \] Now we differentiate \( \cos x \ln x \): Using the product rule: \[ \frac{d}{dx}(\cos x \ln x) = \frac{d}{dx}(\cos x) \cdot \ln x + \cos x \cdot \frac{d}{dx}(\ln x) \] Calculating the derivatives: \[ \frac{d}{dx}(\cos x) = -\sin x \] \[ \frac{d}{dx}(\ln x) = \frac{1}{x} \] Thus, \[ \frac{d}{dx}(\cos x \ln x) = -\sin x \ln x + \cos x \cdot \frac{1}{x} \] Putting it all together: \[ \frac{dz}{dx} = x^{\cos x} \left(-\sin x \ln x + \frac{\cos x}{x}\right) \] Now substituting back into \( \frac{du}{dx} \): \[ \frac{du}{dx} = \cos(x^{\cos x}) \cdot x^{\cos x} \left(-\sin x \ln x + \frac{\cos x}{x}\right) \] 3. **Putting it all together**: Now we can substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule: \[ \frac{dy}{dx} = \frac{x^3 \cdot \left[\cos(x^{\cos x}) \cdot x^{\cos x} \left(-\sin x \ln x + \frac{\cos x}{x}\right)\right] - \sin(x^{\cos x}) \cdot 3x^2}{(x^3)^2} \] Simplifying this gives: \[ \frac{dy}{dx} = \frac{x^3 \cos(x^{\cos x}) x^{\cos x} \left(-\sin x \ln x + \frac{\cos x}{x}\right) - 3x^2 \sin(x^{\cos x})}{x^6} \] Finally, we can simplify the expression: \[ \frac{dy}{dx} = \frac{\cos(x^{\cos x}) x^{\cos x} \left(-\sin x \ln x + \frac{\cos x}{x}\right) - \frac{3 \sin(x^{\cos x})}{x}}{x^3} \] Thus, the derivative \( \frac{dy}{dx} \) is: \[ \frac{dy}{dx} = \frac{\cos(x^{\cos x}) x^{\cos x} \left(-\sin x \ln x + \frac{\cos x}{x}\right) - 3 \sin(x^{\cos x