Suppose f is a differentiable function of x and y, and g(r, s) = f(5r - s, s^2 - 3r)....

Question:

Partial Derivatives; Chain Rule:

The Chain Rule tells us:

{eq}\begin{align*} g_r(r,s)&=f_x(x,y)\cdot x_r(r,s)+f_y(x,y)\cdot y_r(r,s),\\ &\text{and}\\ g_s(r,s)&=f_x(x,y)\cdot x_s(r,s)+f_y(x,y)\cdot y_s(r,s), \end{align*} {/eq}

where

{eq}x=5r - s, \text{ and }y= s^2 - 3r . {/eq}

So for {eq}(r,s)=(2,7) {/eq} we have {eq}(x,y)=(3,43). {/eq}

From the table we need {eq}f_x(3,43) \text{ and }f_y(3,43) {/eq} and by direct calculation we can get the partial derivatives {eq}x_r(2,7), {/eq}

{eq}x_s(2,7), {/eq}

{eq}y_r(2,7), {/eq} and

{eq}y_s(2,7). {/eq}

We evaluate {eq}\ g_r(2, 7) {/eq} and{eq}\ g_s(2, 7), {/eq} by substituting the values of the partial derivatives found above.

Answer and Explanation: 1