Find the gradients of the following functions:

(a) $f(x,y,z)=x$⁴ +$y$³ +$z$⁴

(b)$f(x,y,z)=$$x$²y³z⁴

(c)$f(x,y,z)=e$^x$\sin \left(y\right)In\left(z\right)$

The given functions are,

(a) $f(x,y,z)=x$⁴$+y$³$+z$⁴

(b) $f(x,y,z)=x$²$y$³$z$⁴

(c) $f(x,y,z)=e$^x $\sin \left(y\right)In\left(z\right)$$\sin \left(y\right)In\left(z\right)$

The gradient of the function is defined as its slope on the curve of that function for the given particular point.

Write the given function.

$f(x,y,z)=x$⁴ + $y$³$+z$⁴

Differentiate the above function as,

$\partial f/\partial x=2x$

$\partial f/\partial y=3y$²

$\partial f/\partial z=4z$²

Then, the gradient of the function is written as,

$\nabla f=\partial f/\partial xx+\partial f/\partial yy+\partial f/\partial zz$

$=\left(2x\right)x+(3y$²$)y+(4z$³$)z$

Write the given function.

$f(x,y,z)=x$²$y$³$z$⁴

Differentiate the above function as,

$\partial f/\partial x=2xy$²$z$⁴

$\partial f/\partial y=3x$²$y$²$z$⁴

$\partial f/\partial f=4x$² $y$³$z$³

Then, the gradient of the function is written as,

$\nabla f=\partial f/\partial xx+\partial f/\partial yy+\partial f/\partial zz$

$=(2xy$²$z$⁴$)$$x+(3x$²$y$²$z$⁴$)y+(4x$²y³ z³ $)z$

Write the given function.

$f(x,y,z)=e$^x $\sin \left(y\right)In\left(z\right)$

Differentiate the above function as,

$\partial f/\partial x=e$^x$\sin yInz$

$\frac{df}{dy}=e$^x$\cos yInz$

$\frac{\partial f}{\partial z}=e$^x$\sin y/z$

Then, the gradient of the function is written as,

$\nabla ·f=\frac{\partial f}{\partial x}x+\frac{\partial f}{\partial y}y+\frac{\partial f}{\partial z}z$

$=(e$^x$\sin yInz)x+(e$^x $\cos yInz)$

$y+(e$^x $\sin y/z)z$

Therefore, the gradient of the function is.

$(e$^x$\sin yInz)x+(e$^x$\cos yInz)y+(e$^x$\sin y/z)z$