Partial Derivatives — Question 1

Question 1 – Solution

Strategy. Differentiate term by term, holding the non-differentiated variable fixed.

Step 1: Differentiate with respect to $x$ $$f_x=12x^2y^2-10xy.$$ The terms $7y^3$ and $-6$ are constant with respect to $x$.

Step 2: Differentiate with respect to $y$ $$f_y=8x^3y-5x^2+21y^2.$$

Step 3: Evaluate $$f_x(1,-2)=12(1)(4)-10(1)(-2)=\boxed{68},$$ $$f_y(1,-2)=8(1)(-2)-5(1)+21(4)=\boxed{63}.$$

Verification. The $x$-derivative lowers only powers of $x$; the $y$-derivative lowers only powers of $y$. Substitution gives $48+20=68$ and $-16-5+84=63$.