Partial Derivatives Example 8 Solution

$$\frac{\partial}{\partial x}\cos(xe^{3x}) = -\sin(xe^{3x})\frac{\partial}{\partial x}(xe^{3x}) = -\sin(xe^{3x})\Bigg[e^{3x}\frac{\partial}{\partial x}x+x\frac{\partial}{\partial x}e^{3x}\Bigg] = -\sin(xe^{3x})\Bigg[e^{3x}(1)+x(3e^{3x})\Bigg] = -(1+3x)e^{3x}\sin(xe^{3x})$$

$$\frac{\partial}{\partial y}\cos(xe^{3x}) = -\sin(xe^{3x})\frac{\partial}{\partial y}(xe^{3x}) = -\sin(xe^{3x})\Bigg[e^{3x}\frac{\partial}{\partial y}x+x\frac{\partial}{\partial y}e^{3x}\Bigg] = -\sin(xe^{3x})\Bigg[e^{3x}(0)+x(0)\Bigg] = 0$$