Question

Find the gradients of the following functions:

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

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

(c) $\mathit{f}\left(x,y,z\right)\mathbf{=}{\mathit{e}}^{\mathbf{x}}\mathit{s}\mathit{i}\mathit{n}\left(y\right)\mathit{l}\mathit{n}\left(z\right)$

Short answer

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

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

(c) The gradient of the function is $\left(e^x\sin y\ln z\right)\hat{x}+\left(e^x\cos y\ln z\right)\hat{y}+\left(\frac{e^x\sin y}{z}\right)\hat{z}$.

Step-by-step solution

Write the given information.

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

Define gradient of the function.

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

Solve for the gradient of part (a). 

Write the given function.

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

Differentiate the above function as,

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

Then, the gradient of the function is written as,

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

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

Solve for the gradient of part (b).

Write the given function.

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

Differentiate the above function as,

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

Then, the gradient of the function is written as,

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

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

Solve for the gradient of part (c).

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,

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

Then, the gradient of the function is written as,

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

Therefore, the gradient of the function is.

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