find the indicated partial derivatives of (f(x,y)=7x^{3}y^{2}) using the limit definition. the limits need…

Explanation:

Step1: Find (f(x + h,y))

Given (f(x,y)=7x^{3}y^{2}), then (f(x + h,y)=7(x + h)^{3}y^{2}). Expand ((x + h)^{3}) using the formula ((a + b)^{3}=a^{3}+3a^{2}b + 3ab^{2}+b^{3}), so (f(x + h,y)=7(x^{3}+3x^{2}h + 3xh^{2}+h^{3})y^{2}=7x^{3}y^{2}+21x^{2}hy^{2}+21xh^{2}y^{2}+7h^{3}y^{2}).

Step2: Calculate (f_x(x,y)) using the limit - definition

[ \begin{align*} f_x(x,y)&=\lim_{h\rightarrow0}\frac{f(x + h,y)-f(x,y)}{h}\ &=\lim_{h\rightarrow0}\frac{(7x^{3}y^{2}+21x^{2}hy^{2}+21xh^{2}y^{2}+7h^{3}y^{2})-7x^{3}y^{2}}{h}\ &=\lim_{h\rightarrow0}\frac{21x^{2}hy^{2}+21xh^{2}y^{2}+7h^{3}y^{2}}{h}\ &=\lim_{h\rightarrow0}(21x^{2}y^{2}+21xh y^{2}+7h^{2}y^{2}) \end{align*} ] As (h\rightarrow0), (f_x(x,y)=21x^{2}y^{2}).

Step3: Find (f(x,y + h))

(f(x,y + h)=7x^{3}(y + h)^{2}). Expand ((y + h)^{2}) using the formula ((a + b)^{2}=a^{2}+2ab + b^{2}), so (f(x,y + h)=7x^{3}(y^{2}+2yh+h^{2})=7x^{3}y^{2}+14x^{3}yh + 7x^{3}h^{2}).

Step4: Calculate (f_y(x,y)) using the limit - definition

[ \begin{align*} f_y(x,y)&=\lim_{h\rightarrow0}\frac{f(x,y + h)-f(x,y)}{h}\ &=\lim_{h\rightarrow0}\frac{(7x^{3}y^{2}+14x^{3}yh + 7x^{3}h^{2})-7x^{3}y^{2}}{h}\ &=\lim_{h\rightarrow0}\frac{14x^{3}yh + 7x^{3}h^{2}}{h}\ &=\lim_{h\rightarrow0}(14x^{3}y+7x^{3}h) \end{align*} ] As (h\rightarrow0), (f_y(x,y)=14x^{3}y).

Answer:

(f_x(x,y)=21x^{2}y^{2}), (f_y(x,y)=14x^{3}y)