What is the derivative of xy?

Understand the Problem

The question is asking for the derivative of the product of two variables, x and y, which may depend on a third variable. We will likely use the product rule of differentiation to solve this.

Answer

$$ \frac{d}{dt}(xy) = \frac{dx}{dt} \cdot y + x \cdot \frac{dy}{dt} $$

Steps to Solve

Identify the product rule
Recall that the product rule states that if you have two functions, say $u(x)$ and $v(x)$, their derivative is given by:
$$ \frac{d}{dx}(u \cdot v) = u' \cdot v + u \cdot v' $$
In this case, we will let $u = x$ and $v = y$.
Differentiate each variable
We need to differentiate the variables $x$ and $y$ with respect to the third variable (let's say $t$):

The derivative of $x$ with respect to $t$ is $dx/dt$ (or $x'$).
The derivative of $y$ with respect to $t$ is $dy/dt$ (or $y'$).

Apply the product rule
Now we can apply the product rule we stated earlier:
$$ \frac{d}{dt}(x \cdot y) = \frac{dx}{dt} \cdot y + x \cdot \frac{dy}{dt} $$
Simplify the final expression
Combine the derivatives obtained from the product rule:
$$ \frac{d}{dt}(xy) = x' \cdot y + x \cdot y' $$

This gives us the derivative of the product $xy$ in terms of the derivatives of $x$ and $y$.

$$ \frac{d}{dt}(xy) = \frac{dx}{dt} \cdot y + x \cdot \frac{dy}{dt} $$

More Information

The product rule is an essential part of calculus that allows us to differentiate products of functions. It's useful in various fields such as physics, engineering, and economics, where multiple interdependent variables are common.

Tips

Forgetting to differentiate both functions in the product. Remember to apply the product rule correctly and include both derivatives.
Confusing the order of the terms in the product rule. Make sure to write it as $u'v + uv'$ for functions $u$ and $v$.

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