Exercise · ZW-PARTIAL-DERIVATIVE-009 · ★★★ Advanced
For \(f(x,y)=|x|+y\), which statement is true at the origin?
Choose one option.
Show answer
- B. \(f_x(0,0)\) does not exist and \(f_y(0,0)=1\)
Explanation
With \(y=0\) fixed, \(f(x,0)=|x|\) has a corner at \(x=0\): the one-sided limits of the quotient are \(1\) and \(-1\), so \(f_x(0,0)\) does not exist. On the other hand \(f(0,y)=y\) has derivative \(1\).
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?