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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Step 1: Take the natural logarithm of both sides. Given y = ( x)^ x. Taking the natural logarithm on both sides: y = ( ( x)^ x ) Step 2: Use the logarithm property (a^b) = b a. y = ( x) ( x) Note that x typically refers to _10 x in general contexts, but in calculus, it often implies x (natural logarithm). However, given the explicit use of in the solution, it's safer to assume x means _e x or x. If it were _10 x, the derivative would involve an extra factor of (1)/( 10). Let's proceed assuming x = x. So, the equation becomes: y = ( x) ( x) Step 3: Differentiate both sides with respect to x using the chain rule and product rule. The derivative of y with respect to x is (1)/(y) (dy)/(dx). For the right side, use the product rule (d)/(dx)(uv) = u'v + uv' where u = x and v = ( x). (du)/(dx) = (d)/(dx)( x) = (1)/(x) (dv)/(dx) = (d)/(dx)(( x)) Using the chain rule for v, let w = x. Then v = w. (dv)/(dx) = (d)/(dw)( w) · (dw)/(dx) = (1)/(w) · (1)/(x) = (1)/( x) · (1)/(x) Now apply the product rule: (d)/(dx) (( x) ( x)) = ((1)/(x)) ( x) + ( x) ((1)/(x x)) (d)/(dx) (( x) ( x)) = (( x))/(x) + (1)/(x) Step 4: Equate the derivatives and solve for (dy)/(dx). (1)/(y) (dy)/(dx) = (( x))/(x) + (1)/(x) (1)/(y) (dy)/(dx) = (1 + ( x))/(x) Multiply both sides by y: (dy)/(dx) = y ( (1 + ( x))/(x) ) Substitute y = ( x)^ x back into the equation: (dy)/(dx) = ( x)^ x ( (1 + ( x))/(x) ) The final answer is ( x)^ x ( (1 + ( x))/(x) ). 3 done, 2 left today. You're making progress.