Ini Belog Korie: Differentiation of the Function y=ax, Differentiation of the Sum and Difference of Algebraic Functions

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,

If $n = 1, y = a x$ , 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

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

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

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

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

Thursday, October 29, 2009

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

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