Q. 2 TF - Calculus

Question

A kind of derivative for a function of three variables: Explain why the derivative of the function $$xe^{−4z} \sin{y}$$ is $$e^{−4z} \sin{y}$$  if $$x$$ is the variable and $$y$$ and $$z$$ are constants, and the derivative is $$xe^{−4z} \cos{y}$$  if $$y$$ is the variable and $$x$$ and $$z$$ are constants, and the derivative is $$-4xe^{−4z} \sin{y}$$  if $$z$$ is the variable and $$x$$ and $$y$$ are constants. What is the derivative if $$x$$, $$y$$, and $$z$$ are all constants? 

Step-by-Step Solution

Answer

The derivative when both the variables x and yare kept constants is $$0$$

Step 1. Given Information

Given function is in three variable $$x$$, $$y$$  & $$z$$ and is defined by $$xe^{−4z} \sin{y}$$ 

Step 2. Differentiation when $$x$$ is variable and $$y$$ & $$z$$ are constants

Since $$x$$ is the only variable here, the function which needs to be differentiated is only $$x$$. The rest part of $$xe^{−4z} \sin{y}$$ acts as a constant i.e. $$e^{−4z} \sin{y}$$. Now the given function $$xe^{−4z} \sin{y}$$  is termed as $$kx$$ where  $$k$$ is a constant,$$k= e^{−4z} \sin{y}$$. Now we know the differentiation of $$kx$$ is $$k$$ i.e. the constant part only. Hence, the derivative of the function $$xe^{−4z} \sin{y}$$ is $$e^{−4z} \sin{y}$$  if $$x$$ is the variable and $$y$$ and $$z$$ are constants.

Step 3. Differentiation when $$y$$ is variable and $$x$$ & $$z$$ are constants

Similar to the step 2. Since $$y$$ is the only variable here, the function which needs to be differentiated is only $$\sin{y}$$. The rest part of $$xe^{−4z} \sin{y}$$ acts as a constant i.e. $$xe^{−4z}$$. Now the given function $$xe^{−4z} \sin{y}$$  is termed as $$k\sin{y}$$ where  $$k$$ is a constant,$$k= xe^{−4z} $$. Now we know the differentiation of $$k\sin{y}$$ is $$k\cos{y}$$. Hence, the derivative of the function $$xe^{−4z} \sin{y}$$ is $$xe^{−4z} \cos{y}$$  if $$y$$ is the variable and $$x$$ and $$z$$ are constants.

Step 4. Differentiation when $$z$$ is variable and $$x$$ & $$y$$ are constants

Similar to the step 3. Since $$z$$ is the only variable here, the function which needs to be differentiated is only $$e^{-4z}$$. The rest part of $$xe^{−4z} \sin{y}$$ acts as a constant i.e. $$x\sin{y}$$. Now the given function $$xe^{−4z} \sin{y}$$  is termed as $$ke^{-4z}$$ where  $$k$$ is a constant,$$k= x\sin{y} $$. Now we know the differentiation of $$ke^{-4z}$$ is $$-4ke^{-4z}$$. Hence, the derivative of the function $$xe^{−4z} \sin{y}$$ is $$-4xe^{−4z} \sin{y}$$  if $$z$$ is the variable and $$x$$ and $$y$$ are constants.

Step 5. Differentiation when all the variables are constants i.e. $$x$$, $$y$$ & $$z$$ are constants

Since, in this case all the variables are constants, the given function acts as constant. Hence, we have to differentiate a constant function where constant  $$k$$ (say) is $$xe^{−4z} \sin{y}$$ We know that the differentiation of a constant function is zero. Hence, the differentiation of $$xe^{−4z} \sin{y}$$ is $$0$$ when all the variables are constants.