Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §13.5 The Chain Rule, #40 · Multivariable Calculus · Partial derivatives

(a) For … compute … (no subscript, … all independent)

(a) For $f = x^{2} + y^{2} + z^{2}$ compute $\partial f / \partial x$ (no subscript, $x, y, z$ all independent).
(b) When there is a further relation $z = x^{2} + y^{2}$ , use it to remove $z$ and compute $(\partial f / \partial x)_{y}$ .
(c) Compute $(\partial f / \partial x)_{y}$ using the chain rule $(\partial f / \partial x) + (\partial f / \partial z) (\partial z / \partial x)$ .
(d) Why doesn't that chain rule contain $(\partial f / \partial y) (\partial y / \partial x)$ ?

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §13.5 The Chain Rule, #40

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