Multivariable calculus

Directional derivatives

The rate of change in a chosen direction, not along an axis.

• Directional derivative of x^2 y at (1, 2)The directional derivative of f(x,y) = x^2 y at (1, 2) in the unit direction (0.6, 0.8) is 16/5. Gradient, dot product, and the normalising mistake.
• Directional derivative of e^(xy) at (1, 1)The directional derivative of f(x,y) = e^(xy) at (1, 1) in the unit direction (0.28, 0.96) is 31e/25, about 3.3707. Both partials carry a chain factor.
• Directional derivative of ln(x^2 + y^2) at (1, 2)The directional derivative of f(x,y) = ln(x^2 + y^2) at (1, 2) in the unit direction (0.8, -0.6) is -4/25 = -0.16, so f decreases that way.
• Directional derivative of sin(xy) at (0, 2)The directional derivative of f(x,y) = sin(xy) at (0, 2) in the unit direction (0.6, -0.8) is 6/5 = 1.2. The whole answer comes from the x component.
• Directional derivative of x^3 - 3xy^2 at (2, 1)The directional derivative of f(x,y) = x^3 - 3xy^2 at (2, 1) in the unit direction (0.8, 0.6) is 0, because that direction is tangent to the level curve.
• Directional derivative of sqrt(x^2 + y^2) at (3, 4)The directional derivative of f(x,y) = sqrt(x^2 + y^2) at (3, 4) in the unit direction (0.352, 0.936) is 24/25 = 0.96, the cosine of the angle involved.
• Directional derivative of arctan(y/x) at (1, 1)The directional derivative of f(x,y) = arctan(y/x) at (1, 1) in the unit direction (-0.6, 0.8) is 7/10 = 0.7. The gradient circles the origin.
• Directional derivative of x/y at (2, 1)The directional derivative of f(x,y) = x/y at (2, 1) in the unit direction (0.96, 0.28) is 2/5 = 0.4. The y partial needs the power rule on 1/y.
• Directional derivative of e^(2x - y) at (1, 2)The directional derivative of f(x,y) = e^(2x - y) at (1, 2) in the unit direction (-0.8, 0.6) is -11/5 = -2.2, close to the steepest possible descent.
• Directional derivative of x^2 + 4y^2 at (1, 1)The directional derivative of f(x,y) = x^2 + 4y^2 at (1, 1) in the unit direction (0.96, -0.28) is -8/25 = -0.32, just past the tangent to the level ellipse.
• Directional derivative of x e^y at (3, 0) along (0, 1)The directional derivative of f(x,y) = x e^y at (3, 0) in the unit direction (0, 1) is 3. Along an axis direction, the answer is just a partial derivative.
• Directional derivative of x^3 + y^3 - 3xy at (1, 2)The directional derivative of f(x,y) = x^3 + y^3 - 3xy at (1, 2) in the unit direction (-0.28, 0.96) is 237/25 = 9.48, near the steepest rate available.