Find The First Derivative of y = x ^x 先存在这里

Find The First Derivative of y = x ^x

 

Note that the function defined by y = x ^x is neither a power function of the form x ^k nor an exponential function of the form b ^x and the formulas of Differentiation of these functions cannot be used. We need to find another method to find the first derivative of the above function.

If y = x ^x and x > 0 then ln y = ln (x ^x)

Use properties of logarithmic functions to expand the right side of the above equation as follows.

ln y = x ln x

We now differentiate both sides with respect to x, using chain rule on the left side and the product rule on the right.

y '(1 / y) = ln x + x(1 / x) = ln x + 1 , where y ' = dy/dx

Multiply both sides by y

y ' = (ln x + 1)y

Substitute y by x ^x to obtain

y ' = (ln x + 1)x ^x

 

Exercise: Find the first derivative of y = x^{x - 2}

Answer to Above Exercise: y ' = x ^{x - 3} (x ln x + x - 2)