Answer to Question #138346 in Calculus for Promise Omiponle

Question

If f(x,y,z)=xe^(2y)sin(9z), then the gradient is

∇f(x,y,z)=

Answer 1

$\displaystyle \nabla f(x, y, z) = \frac{\partial f(x, y, z)}{\partial x} \textbf{i} + \frac{\partial f(x, y, z)}{\partial y} \textbf{j} + \frac{\partial f(x, y, z)}{\partial z} \textbf{k} \\ \frac{\partial f(x, y, z)}{\partial x} = \frac{\partial (xe^{2y}\sin(9z))}{\partial x} = e^{2y}\sin(9z) \\ \frac{\partial f(x, y, z)}{\partial y} = \frac{\partial (xe^{2y}\sin(9z))}{\partial y} = 2xe^{2y}\sin(9z)\\ \frac{\partial f(x, y, z)}{\partial z} = \frac{\partial (xe^{2y}\sin(9z))}{\partial z} = 9xe^{2y}\cos(9z)\\ \therefore \nabla f(x, y, z) = e^{2y}\sin(9z) \textbf{i} + 2xe^{2y}\sin(9z)\textbf{j} + 9xe^{2y}\cos(9z)\textbf{k} \\ \therefore \nabla f(x, y, z) = e^{2y}(\sin(9z) \textbf{i} + 2x\sin(9z)\textbf{j} + 9x\cos(9z)\textbf{k})\\ \implies \nabla f(x, y, z) = e^{2y}(\sin(9z), 2x\sin(9z), 9x\cos(9z))$