Differentiation of the Function y=ax, Differentiation of the Sum and Difference of Algebraic Functions

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In general,

If \, y = ax^n then \frac{{dy}}{{dx}} = ax^{n - 1} or \frac{{\rm d}}{{{\rm dx}}}(ax^n ) = anx^{n - 1} where a is a constant and n is an integer and rational number.

In particular,

Ifn = 1,y = ax, then \frac{{dy}}{{dx}} = a

If n = 0,y = a, then \frac{{dy}}{{dx}} = 0

Example

Find the \frac{{dy}}{{dx}} when y=4\sqrt x + 3x^4 - \frac{{5}}{{x^3}}

Solution

y=4\sqrt x + 3x^4 - \frac{{5}}{{x^3}}

=4x^\frac{{1}}{{2}} + 3x^4 - 5x^{-3}

\frac{{dy}}{{dx}}= \frac {{1}}{{2}}(4x^{-\frac{{1}}{{2}}})+4(3x^3)-(-3)(5x^{-4})

= \frac {{2}}{{\sqrt x}} + 12x^3 + \frac {{15}}{{x^4}}

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