Question

If \(f\left( {x,y} \right) = \sqrt(3){{{x^3} + {y^3}}}\), find \({f_x}\left( {0,0} \right)\).

Short answer

A limit is the value that a function approaches as the input approaches a certain value in mathematics. Limits are used to define continuity, derivatives in calculus and mathematical analysis.

Step-by-step solution

Given Data,

Given: \(f\left( {x,y} \right) = \sqrt(3){{{x^3} + {y^3}}}\)

Partial Derivative.

\(\begin{aligned}{c}{f_x}\left( {0,0} \right) = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {0 + h,0} \right) - f\left( {0,0} \right)}}{h}\\ = \mathop {\lim }\limits_{h \to 0} \frac{{\sqrt(3){{{{\left( {0 + h} \right)}^3} + 0}} - 0}}{h}\\ = \mathop {\lim }\limits_{h \to 0} \frac{h}{h}\\ = \mathop {\lim }\limits_{h \to 0} 1\\ = 1\end{aligned}\)

Hence, \({f_x}\left( {0,0} \right) = 1\).