differentiate y=x^2x. show all work
To differentiate the expression y = x^2x, you can use the product rule and the power rule of differentiation. Here's how you can break it down step by step:
Step 1: Rewrite the expression using the exponent rule.
y = x^(2x)
Step 2: Take the natural logarithm (ln) of both sides of the equation.
ln(y) = ln(x^(2x))
Step 3: Apply the logarithm rule to remove the exponent.
ln(y) = (2x) * ln(x)
Step 4: Differentiate both sides of the equation implicitly with respect to x.
d/dx(ln(y)) = d/dx((2x) * ln(x))
Step 5: Apply the chain rule to the left-hand side.
(1/y) * (dy/dx) = 2 * ln(x) + 2x * (1/x)
Step 6: Simplify the expression on the right-hand side.
(dy/dx) / y = 2 * ln(x) + 2
Step 7: Multiply both sides of the equation by y.
dy/dx = y * (2 * ln(x) + 2)
Step 8: Substitute back the value of y.
dy/dx = x^(2x) * (2 * ln(x) + 2)
That's it! You have differentiated the expression y = x^2x using the product rule and the power rule. The final answer is dy/dx = x^(2x) * (2 * ln(x) + 2).