Find derivatives of following functions . y=a^x(agt 0,aneq 1) y=x^x

## Step 1:To find the derivative of the function , where and , we use the rule of differentiation for exponential functions.### **The derivative of with respect to is \(a^x \ln(a)\)**This rule applies because is a constant and is the variable.## Step 2:For the function , we use the chain rule and the property of logarithmic differentiation. This is because both the base and the exponent are variables.## Step 3:We first transform the function by taking the natural logarithm of both sides, which gives us \(\ln(y) = \ln(x^x)\).## Step 4:Simplifying the right side, we get \(\ln(y) = x \ln(x)\).## Step 5:Next, we differentiate both sides with respect to and use the product rule on the right side, which gives us \(\frac{1}{y} \cdot \frac{dy}{dx} = \ln(x) + x \cdot \frac{1}{x}\).## Step 6:Solving for , we find \(\frac{dy}{dx} = y \cdot (\ln(x) + 1)\).## Step 7:Finally, we substitute back , which gives us \(\frac{dy}{dx} = x^x (\ln(x) + 1)\).