What is the derivative of arctan(xy)?
Understand the Problem
The question is asking for the derivative of the function arctan(xy) with respect to an implied variable. This implies using the chain rule and product rule to differentiate the arctangent function in relation to its arguments, which are functions involving x and y.
Answer
The derivative is \( \frac{dz}{dx} = \frac{y}{1 + (xy)^2} \).
Answer for screen readers
The derivative of ( z = \arctan(xy) ) with respect to ( x ) is given by:
$$ \frac{dz}{dx} = \frac{y}{1 + (xy)^2} $$
Steps to Solve
- Identify the function to differentiate
The function we need to differentiate is $z = \arctan(xy)$. Here, $z$ is a function of the product of $x$ and $y$.
- Apply the chain rule
To find the derivative $dz/dx$, we need to use the chain rule. The derivative of $\arctan(u)$, where $u = xy$, is given by:
$$ \frac{dz}{du} = \frac{1}{1 + u^2} $$
- Find the derivative of the inner function
Next, we need to differentiate $u = xy$ with respect to $x$. By applying the product rule, we have:
$$ \frac{du}{dx} = y $$
- Combine using the chain rule
Now, we can combine the results from step 2 and step 3 using the chain rule:
$$ \frac{dz}{dx} = \frac{dz}{du} \cdot \frac{du}{dx} = \frac{1}{1 + (xy)^2} \cdot y $$
- Final formulation
So, the final expression for the derivative of $z$ with respect to $x$ is:
$$ \frac{dz}{dx} = \frac{y}{1 + (xy)^2} $$
The derivative of ( z = \arctan(xy) ) with respect to ( x ) is given by:
$$ \frac{dz}{dx} = \frac{y}{1 + (xy)^2} $$
More Information
The derivative of the arctangent function is often useful in various applications, including calculus and physics, especially when dealing with angles or slopes in relation to coordinate changes.
Tips
- Forget to use the chain rule: It's important to remember to use the chain rule when differentiating functions that are compositions of other functions.
- Neglecting the product rule: When differentiating products of functions, like ( xy ), it's vital to apply the product rule correctly.
