This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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i) Differentiate x^e + x^x + e^x + e^e with respect to x. Step 1: Differentiate each term separately. We use the power rule (d)/(dx)(x^n) = nx^n-1, the derivative of e^x is e^x, and the derivative of a constant is 0. For x^x, we use logarithmic differentiation. Let y_1 = x^x. Then y_1 = x x. Differentiating with respect to x: (1)/(y_1) (dy_1)/(dx) = (d)/(dx)(x) · x + x · (d)/(dx)( x) (1)/(y_1) (dy_1)/(dx) = 1 · x + x · (1)/(x) (1)/(y_1) (dy_1)/(dx) = x + 1 (dy_1)/(dx) = y_1 (1 + x) = x^x (1 + x) Step 2: Differentiate x^e. Using the power rule (d)/(dx)(x^n) = nx^n-1: (d)/(dx)(x^e) = e x^e-1 Step 3: Differentiate e^x. (d)/(dx)(e^x) = e^x Step 4: Differentiate e^e. Since e^e is a constant: (d)/(dx)(e^e) = 0 Step 5: Combine the derivatives. (d)/(dx)(x^e + x^x + e^x + e^e) = (d)/(dx)(x^e) + (d)/(dx)(x^x) + (d)/(dx)(e^x) + (d)/(dx)(e^e) (d)/(dx)(x^e + x^x + e^x + e^e) = e x^e-1 + x^x (1 + x) + e^x + 0 The derivative is: e x^e-1 + x^x (1 + x) + e^x ii) Differentiate ( x)^ x + ( x)^ x with respect to x. Let $y = ( x)^ x + ( x)^ x