Q. 37 - Calculus

Question

Find the gradient of the given functions in Exercises 37–42.

$z=x^2\sin y+y\sin x$

Step-by-Step Solution

Answer

The gradient of the given function is$\left(2x\sin y+y\cos x, x^2\cos y+\sin x\right)$.

Step 1. Given Information.

The given function is:

$z=x^2\sin y+y\sin x$

Step 2. Calculation.

The gradient of the given function is:

$$\begin{gathered} z=f\left(x,y\right)=x^2\sin y+y\sin x \\ \nabla f(x,y)=\left[f_x\left(x,y\right),f_y\left(x,y\right)\right]-------\left(1\right) \end{gathered}$$

Now find

$$\begin{gathered} f_x\left(x,y\right)=2x\sin y+y\cos x \\ {f}_y\left(x,y\right)=x^2\cos y+\sin x \end{gathered}$$

Use these above values in (1) we get,

$\nabla f\left(x,y\right)=\left(2x\sin y+y\cos x, x^2\cos y+\sin x\right)$

Step 3. Conclusion.

The gradient of the given function is $\left(2x\sin y+y\cos x, x^2\cos y+\sin x\right)$.