Topics:

• More on the Chain Rule and the two special cases

Here is a revised version of the slideshow which contains the two special cases and also information about implicit differentiation:

MAT1475CompositeFnsChainRule-slideshow

• Examples using the General Power Rule:

$f(x) = \sqrt{4-x^{2}}$

$f(x) = \frac{1}{x^{2} + 5}$

• Examples using the Exponential Function Rule:

$f(x) = e^{3x}$ (was done last time)

$f(x) = e^{-x^{2}}$

• Using the Chain Rule along with the Product Rule

• Finding the equation of the tangent line to the graph of $y = \sqrt{4-x}$ at the point $(3,1)$

• Introduction to implicit differentiation: we assume y is a function of x and use the Chain Rule in the form $\frac{\textrm{d}}{\textrm{d}x}f(y) = \frac{\textrm{d}f}{\textrm{d}y}\frac{\textrm{d}y}{\textrm{d}x}$

Some examples:

$\frac{\textrm{d}}{\textrm{d}x}\left(y^{2}\right) = 2y\frac{\textrm{d}y}{\textrm{d}x}$

$\frac{\textrm{d}}{\textrm{d}x}\left(x^{3}y\right) = x^{3}\frac{\textrm{d}y}{\textrm{d}x} + 3x^{2}y$ (Use the Product Rule along with the Chain Rule.)

Then we found the slope of the tangent line to the graph $x^{2} + y^{2} = 16$, $y\ge 0$, at the point where $x = 1$.