3.3. DERIVATIVES247Finally, we note that, as a corollary of Proposition3.3.7, we get a formulafor the partial derivatives of the composite functiong◦f:∂g◦f∂xi(−→x0) =g′(y0)∂f∂xi(−→x0).(3.11)For example, suppose we consider the function that expresses therectangular coordinateyin terms of spherical coordinates:f(ρ,φ,θ) =ρsinφsinθ;its gradient is−→∇f(ρ,φ,θ) = (sinφsinθ,ρcosφsinθ,ρsinφcosθ).Suppose further that we are interested inz=g(y)= lny.
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