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Exercise 524Calculus › Multivariable calculus › Partial derivativesHigh schoolAdded 3 Nov 2026

The gradient

#gradient#vector#direction

Statement

Let f(x,y,z)=x2+y2+z2f(x,y,z)=x^2+y^2+z^2. Compute the gradient ∇f\nabla f and evaluate it at (1,2,2)(1,2,2).

Hint

∇f=(∂xf,∂yf,∂zf)\nabla f=(\partial_x f,\partial_y f,\partial_z f).

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Solution

Gradient. ∇f(x,y,z)=(2x,2y,2z)\nabla f(x,y,z)=(2x,2y,2z).

At (1,2,2)(1,2,2). ∇f(1,2,2)=(2,4,4)\nabla f(1,2,2)=(2,4,4).

Interpretation. The gradient points in the direction of steepest increase of ff, with magnitude ∥∇f∥=4+16+16=6\|\nabla f\|=\sqrt{4+16+16}=6.

Conclusion. ∇f(1,2,2)=(2,4,4)\nabla f(1,2,2)=(2,4,4).

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