9.4: Summary

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The chain rule can be used to relate the changes in variables that depend on one other in some "chain of relationships". We use the term "related rates" to describe such problems.
Curves that fail the vertical line property cannot be describe by a single function $y=f(x)$, even if we can represent some of those curves by equation(s).
Zooming in on such a curve, we can define an implicit function that describes some local piece of the curve.
When we use implicit differentiation in two variables, we treat one variable as independent and the other as dependent. This allows us to differentiate the equation with respect to the independent variable using the chain rule.
Through implicit differentiation we showed that the derivative of $y=x^{m / n}$ is $\frac{d y}{d x}=\frac{m}{n} x^{\left(\frac{m}{n}-1\right)}$.
Applications addressed in this chapter included:
1. tumor growth (volume depends on radius which depends on a growth rate);
2. convergent extension in tissue of an embryo (relationship between the length and width of the growing tissue);
3. growth of a cell (the relationship between volume and radius).
4. spider’s thread (length of thread depends on the spider’s position, which depends on time);
5. growth of a tree trunk (determining the fraction of the trunk that is living tissue as the tree grows)
6. conical cup leaking water (height of water depends on volume, which depends on time); and
7. rate of Earth’s temperature change per unit energy loss.

Let $a=b^{2 / 9}$. Determine $\frac{d a}{d b}$.
In Example 9.4, suppose that the cone does not leak, but that it is being filled with water at a constant rates. How would your work change?
Use implicit differentiation to find the slope of the tangent line to the circle $x^{2}+y^{2}=1$ at the point $x=-1$ ? How does your result relate to the orientation of the tangent line to the circle at that point?
Consider the following curve.

$clipboard_e7c70f2f4ee3fb860c7e793c923694402.png$

Draw both tangent lines to this curve at $x=0.5$.

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