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Fractional Indices

The rule for differentiation for a standard function y = xn also works when the function has rational indices, i.e. n is a rational number (either a fraction or decimal number)

{d/dx}x^n  =  nx^{n-1}

 

Let us look at a few examples of fractional indices:

Example 1: y = 4 root{4}{x}

Write y = 4 (x)^{1/4}, and then apply {d/dx}x^n

{dy/dx} = 4 {d/dx}(x)^{1/4}

= 4 x 1/4 (x)^{{1/4}-1}

= 1 x (x)^{-3/4}

= 1/{(x)^{3/4}}

= 1/{root{4}{x^3}}

 

Example 2: y = 1/sqrt{x}

Write y = (sqrt{x})^-1  =  {x}^{-1/2}

{dy/dx}{d/{dx}}{x}^{-1/2}  =  {-1/2}~{x}^{-{1/2}-1}

{-1/2}~{x}^{-{3/2}}  =  -1/{2x^{3/2}}  =  -1/{2 sqrt{x^3}}

 

Example 3: y  =  3/root{3}{x^4}

=  3/{x^{4/3}}  =  3{x^{-4/3}}

{dy/dx}  =  3({-4/3}{x}^{-{4/3}-1})  =  {-4}{x}^{-7/3}

=  {-4}/{x}^{7/3}  =  {-4}/root{3}{x^7}

 

Example 4: Find the equation of tangent to the curve y = {2}/root{3}{x^2} at the point (1, 2)

y  =  2/{x^{2/3}}  =  2{x^{-2/3}}

{dy/dx}  = 2 x -2/3{x^{-2/3-1}}

{-4/3}{x^{-5/3}}

At x = 1, m = {dy/dx}  =  {-4/3}~*~{1^{-5/3}}

=  -4/3

Equation of tangent at (1, 2) is

y – 2 = -4/3(x – 1)

3y – 6 = -4x + 4

4x + 3y – 10 = 0