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प्रश्न
Find the nth order derivative of log x.
बेरीज
उत्तर
Let y = log x
Differentiate w. r. t. x
`dy/dx = d/dx(log x) = 1/x`
Differentiate w. r. t. x
`d/dx (dy/dx) = d/dx(1/x)`
`(d^2y)/(dx^2) = (-1)/x^2 = (-1)^1/x^2`
Differentiate w. r. t. x
`d/dx((d^2y)/(dx^2)) = (-1)^1d/dx(1/x^2)`
`(d^3y)/(dx^3) = (-1)^1((-2)/x^3) = ((-1)^2 * 1 * 2)/x^3`
In general, the nth order derivative will be
`(d^ny)/(dx^n) = ((-1)^(n - 1)*1*2*3... (n - 1))/x^n`
`(d^ny)/(dx^n) = ((-1)^(n - 1)* (n - 1)!)/x^n`
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