91Ó°ÊÓ

The given functions are,

(a) $f\left(x,y,z\right)=x^4+y^3+z^4$

(b) $f\left(x,y,z\right)=x^2{y}^3{z}^4$

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

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.

$f\left(x,y,z\right)=x^4+y^3+z^4$

Differentiate the above function as,

$$\begin{gathered} \frac{∂f}{∂x}=2x \\ \frac{∂f}{∂y}=3y^2 \\ \frac{∂f}{∂z}=4z^3 \end{gathered}$$

Then, the gradient of the function is written as,

$\begin{array}{rcl}∇f& =& \frac{∂f}{∂x}\hat{x}+\frac{∂f}{∂y}\hat{y}+\frac{∂f}{∂z}\hat{z}\\ & =& \left(2x\right)\hat{x}+\left(3y^2\right)\hat{y}+\left(4z^3\right)\hat{z}\\ & & \end{array}$

Therefore, the gradient of the function is$\left(2x\right)\hat{x}+\left(3y^2\right)\hat{y}+\left(4z^3\right)\hat{z}$ .

Write the given function.

$f\left(x,y,z\right)=x^2{y}^3{z}^4$

Differentiate the above function as,

$$\begin{gathered} \frac{∂f}{∂x}=2x{y}^2{z}^4 \\ \frac{∂f}{∂y}=3x^2{y}^2{z}^4 \\ \frac{∂f}{∂z}=4x^2{y}^3{z}^3 \end{gathered}$$

Then, the gradient of the function is written as,

$\begin{array}{rcl}∇f& =& \frac{∂f}{∂x}\hat{x}+\frac{∂f}{∂y}\hat{y}+\frac{∂f}{∂z}\hat{z}\\ & =& \left(2x{y}^2{z}^4\right)\hat{x}+\left(3x^2{y}^2{z}^4\right)\hat{y}+\left(4x^2{y}^3{z}^3\right)\hat{z}\\ & & \end{array}$

Therefore, the gradient of the function is $\left(2x\right)\hat{x}+\left(3y^2\right)\hat{y}+\left(4z^3\right)\hat{z}$.

Write the given function.

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

Differentiate the above function as,

$$\begin{gathered} \frac{∂f}{∂x}=e^x\sin y\ln z \\ \frac{∂f}{∂y}=e^x\cos y\ln z \\ \frac{∂f}{∂z}=\frac{e^x\sin y}{z} \end{gathered}$$

Then, the gradient of the function is written as,

$\begin{array}{rcl}∇f& =& \frac{∂f}{∂x}\hat{x}+\frac{∂f}{∂y}\hat{y}+\frac{∂f}{∂z}\hat{z}\\ & =& \left(e^x\sin y\ln z\right)\hat{x}+\left(e^x\cos y\ln z\right)\hat{y}+\left(\frac{e^x\sin y}{z}\right)\hat{z}\\ & & \end{array}$

Therefore, the gradient of the function is.

$\left(e^x\sin y\ln z\right)\hat{x}+\left(e^x\cos y\ln z\right)\hat{y}+\left(\frac{e^x\sin y}{z}\right)\hat{z}$