Differentiation

For y = x^x, take log of both sides first, d/dx(log y) = (1/y) × (dy/dx), For y = f(x)^g(x), use logarithmic differentiation, log(m/n) = log m - log n

Logarithmic differentiation is used when the variable appears in both base and exponent. The key formula is d/dx(log y) = (1/y) × (dy/dx). Also, log(m/n) = log m - log n, but log(mn) = log m + log n, not multiplication.

-x/y

For implicit differentiation of x² + y² = 25: Differentiate both sides with respect to x: 2x + 2y(dy/dx) = 0. Solving for dy/dx: 2y(dy/dx) = -2x, therefore dy/dx = -x/y

1/t

For parametric functions: dx/dt = 6t and dy/dt = 6. Using the formula dy/dx = (dy/dt)/(dx/dt) = 6/(6t) = 1/t

12x²

First derivative: dy/dx = 4x³. Second derivative: d²y/dx² = d/dx(4x³) = 12x². This is the second-order derivative of the given function.