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Engineering Calculus Notes 259

Engineering Calculus Notes 259 - 247 3.3 DERIVATIVES...

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3.3. DERIVATIVES 247 Finally, we note that, as a corollary of Proposition 3.3.7 , we get a formula for the partial derivatives of the composite function g f : ∂g f ∂x i ( −→ x 0 ) = g ( y 0 ) ∂f ∂x i ( −→ x 0 ) . (3.11) For example, suppose we consider the function that expresses the rectangular coordinate y in terms of spherical coordinates: f ( ρ,φ,θ ) = ρ sin φ sin θ ; its gradient is −→ f ( ρ,φ,θ ) = (sin φ sin θ,ρ cos φ sin θ,ρ sin φ cos θ ) . Suppose further that we are interested in z = g ( y ) = ln y.
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