91Ó°ÊÓ

To understand implicit differentiation, we must grasp the concept of the Chain Rule. The Chain Rule is essential whenever you're differentiating compositions of functions. In simple terms, it helps you differentiate a function that is nested within another function.
When we have an equation like \( \sin xy = x + y \), we notice "\( xy \)" is inside the sine function, which means we need to apply the Chain Rule.
Here's a quick breakdown: In our given equation, the outer function is \( \sin(u) \) and the inner function \( u = xy \). When differentiating \( \sin u \) with respect to \( x \), the chain rule requires us to multiply the derivative of the outer function \( \cos(u) \) by the derivative of the inner function \( \frac{du}{dx} \).
Thus, we get \[ \cos(xy) \cdot \frac{d(xy)}{dx} \]. It’s a way to 'chain' different parts of a composite function together for derivation purposes.
This technique is powerful when working with implicit differentiation where functions of \( y \) are involved.

The Product Rule is another key tool in differentiation, particularly useful when dealing with products of functions. In the context of our exercise, we find \( xy \) within our equation \( \sin xy = x + y \). Here, \( xy \) is a product of \( x \) and \( y \), which requires us to use the Product Rule.
The Product Rule states that when differentiating a product of two functions, say \( u(x) \cdot v(x) \), the formula is \( u'v + uv' \).
Applying this to \( x \cdot y \), gives \( \frac{d}{dx}(xy) = \frac{d}{dx}(x)y + x\frac{d}{dx}(y) = y + x\frac{dy}{dx} \).
This derivative then feeds into the chain rule from the previous section, allowing us to further differentiate more complex parts of the equation.
The Product Rule makes finding derivatives of multiplicative expressions straightforward and systematic, especially when dealing with terms that involve products of variables.

Differentiation is a staple method in calculus used to find the rate at which one quantity changes with respect to another. There are numerous techniques for differentiation, especially useful for finding derivatives in implicit and explicit forms.
Some key techniques used here include the Chain Rule and Product Rule. However, implicit differentiation particularly shines when dealing with functions not easily solved for one variable. When \( y \) is not isolated, like in our example \( \sin xy = x + y \), implicit differentiation allows us to differentiate both sides with respect to \( x \) without explicitly solving for \( y \).
Start by differentiating every term with respect to \( x \). Then, whenever \( y \) appears, multiply its derivative by \( \frac{dy}{dx} \) since \( y \) is considered a function of \( x \).
Finally, after using these techniques, we often end up with a mix of \( y \) and \( \frac{dy}{dx} \). The next step is to rearrange to solve for \( \frac{dy}{dx} \), giving us the needed derivative even when \( y \) is not separately defined.
Implicit differentiation and its associated techniques make handling complex equations much more manageable.