Problem 44 - Calculus Early Transcendentals

Question

Find the first partial derivatives of the following functions. $$Q(x, y, z)=\tan x y z$$

Step-by-Step Solution

Answer

Question: Find the first partial derivatives of the function $$Q(x, y, z)=\tan x y z$$ with respect to x, y, and z. Answer: The first partial derivatives of the function Q(x, y, z) are: $$\frac{\partial Q}{\partial x}=\sec^2(xyz) \cdot yz$$ $$\frac{\partial Q}{\partial y}=\sec^2(xyz) \cdot xz$$ $$\frac{\partial Q}{\partial z}=\sec^2(xyz) \cdot xy$$

Step 1: Write down the given function

The function we are given is $$Q(x, y, z)=\tan x y z$$, and we want to find its partial derivatives with respect to each variable x, y, and z.

Step 2: Differentiate the function with respect to x

We'll start with finding the partial derivative with respect to x. Keep y and z as constant and differentiate the function concerning x: $$\frac{\partial Q}{\partial x}=\frac{\partial}{\partial x}\tan (xyz)$$ Now apply the chain rule. The derivative of $$\tan u$$ concerning $$u$$ is $$\sec^2 u$$ and the derivative of $$xyz$$ regarding $$x$$ is $$yz$$, so the chain rule suggests: $$\frac{\partial Q}{\partial x}=\sec^2(xyz) \cdot yz$$

Step 3: Differentiate the function concerning y

Now let's find the partial derivative with respect to y. Keep x and z as constants and differentiate the function concerning y: $$\frac{\partial Q}{\partial y}=\frac{\partial}{\partial y}\tan (xyz)$$ As we did in the previous step, apply the chain rule, but this time, the derivative of $$xyz$$ regarding $$y$$ is $$xz$$, so the chain rule suggests: $$\frac{\partial Q}{\partial y}=\sec^2(xyz) \cdot xz$$

Step 4: Differentiate the function concerning z

Finally, find the partial derivative with respect to z. Keep x and y as fixed and differentiate the function concerning z: $$\frac{\partial Q}{\partial z}=\frac{\partial}{\partial z}\tan (xyz)$$ Similar to the previous steps, apply the chain rule. The derivative of $$xyz$$ regarding $$z$$ is $$xy$$, so the chain rule suggests: $$\frac{\partial Q}{\partial z}=\sec^2(xyz) \cdot xy$$

Step 5: Write down the results

We have found the first partial derivatives of the function Q(x, y, z) with respect to each variable. They are: $$\frac{\partial Q}{\partial x}=\sec^2(xyz) \cdot yz$$ $$\frac{\partial Q}{\partial y}=\sec^2(xyz) \cdot xz$$ $$\frac{\partial Q}{\partial z}=\sec^2(xyz) \cdot xy$$