Q9:

Differentiate the functions w.r.t. $x$. $x^{\sin x} + (\sin x)^{\cos x}$

Step 1: Decomposition of the Function

Given the function $y = x^{\sin x} + (\sin x)^{\cos x}$. Let $u = x^{\sin x}$ and $v = (\sin x)^{\cos x}$. The function becomes $y = u + v$. To find the derivative $\frac{dy}{dx}$, we must compute $\frac{du}{dx} + \frac{dv}{dx}$ [By the linearity of the derivative operator].

Step 2: Differentiating $u = x^{\sin x}$

Taking the natural logarithm on both sides: $\ln(u) = \ln(x^{\sin x})$. Using the logarithmic property $\ln(a^b) = b \ln a$:

$\ln(u) = \sin x \cdot \ln x$

Differentiating both sides with respect to $x$ [Using the Chain Rule and Product Rule]:

$\frac{1}{u} \frac{du}{dx} = \frac{d}{dx}(\sin x) \cdot \ln x + \sin x \cdot \frac{d}{dx}(\ln x)$

$\frac{1}{u} \frac{du}{dx} = \cos x \cdot \ln x + \sin x \cdot \frac{1}{x}$

$\frac{du}{dx} = u \left( \cos x \ln x + \frac{\sin x}{x} \right) = x^{\sin x} \left( \cos x \ln x + \frac{\sin x}{x} \right)$

Step 3: Differentiating $v = (\sin x)^{\cos x}$

Taking the natural logarithm on both sides: $\ln(v) = \ln((\sin x)^{\cos x})$.

$\ln(v) = \cos x \cdot \ln(\sin x)$

Differentiating both sides with respect to $x$ [Using the Chain Rule and Product Rule]:

$\frac{1}{v} \frac{dv}{dx} = \frac{d}{dx}(\cos x) \cdot \ln(\sin x) + \cos x \cdot \frac{d}{dx}(\ln(\sin x))$

Applying the Chain Rule to $\frac{d}{dx}(\ln(\sin x)) = \frac{1}{\sin x} \cdot \cos x = \cot x$:

$\frac{1}{v} \frac{dv}{dx} = -\sin x \cdot \ln(\sin x) + \cos x \cdot \cot x$

$\frac{dv}{dx} = v \left( \cos x \cot x - \sin x \ln(\sin x) \right) = (\sin x)^{\cos x} \left( \cos x \cot x - \sin x \ln(\sin x) \right)$

Step 4: Combining the Results

Since $\frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx}$, we aggregate the expressions obtained in Step 2 and Step 3:

$\frac{dy}{dx} = x^{\sin x} \left( \cos x \ln x + \frac{\sin x}{x} \right) + (\sin x)^{\cos x} \left( \cos x \cot x - \sin x \ln(\sin x) \right)$

Visual Representation of Derivative Decomposition

Final Solution: \frac{dy}{dx} = x^{\sin x} \left( \cos x \ln x + \frac{\sin x}{x} \right) + (\sin x)^{\cos x} \left( \cos x \cot x - \sin x \ln(\sin x) \right)