Sunshine Maths

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)

${d/dx}x^n$ = n $x^{n-1}$

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

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