Exercise · ZW-PARTIAL-DERIVATIVE-010 · ★★★ Advanced
Which of these functions satisfies \(f_{xx}+f_{yy}=0\) at every point?
Choose one option.
Show answer
- B. \(x^{2} - y^{2}\)
Explanation
For \(x^2-y^2\): \(f_{xx}=2\) and \(f_{yy}=-2\), so the sum is \(0\). With \(x^2+y^2\) it gives \(4\); with \(x^3+y^3\) it gives \(6x+6y\); with \(e^x+y\) it gives \(e^x\).
Review the concept: Partial derivative
The partial derivative of a function \(f(x,y)\) with respect to \(x\) at a point \((a,b)\) is the ordinary derivative of the one-variable function obtained by fixing \(y=b\) and letting only \(x\) vary.
\[f_x(a,b)=\lim_{h\to0}\frac{f(a+h,b)-f(a,b)}{h}\]
More practice problems: partial derivative
- ZW-PARTIAL-DERIVATIVE-001 For f(x,y)=x^2y+3y^3, compute f_x(x,y).
- ZW-PARTIAL-DERIVATIVE-002 For the same function f(x,y)=x^2y+3y^3, compute f_y(x,y).
- ZW-PARTIAL-DERIVATIVE-003 Compute (∂)/(∂ x)sin (xy).
- ZW-PARTIAL-DERIVATIVE-004 If f(x,y)=e^xy^2, what is f_y(1,1)?
- ZW-PARTIAL-DERIVATIVE-005 For f(x,y)=√x^2+y^2, what is f_x(3,4)?
- ZW-PARTIAL-DERIVATIVE-006 If f(x,y)=xln y, what is f_y(2,e)?
- ZW-PARTIAL-DERIVATIVE-007 For f(x,y)=x^2y^3, compute the mixed partial derivative f_xy=(∂)/(∂ y)(f_x).
- ZW-PARTIAL-DERIVATIVE-008 The surface z=9-x^2-y^2 is cut by the plane y=2. What is the slope of the resulting curve where x=1?