Chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions $f$ and $g$ in terms of the derivatives $f$ and $g$. More precisely, if $h=f \circ g$ is the function such that $h(x)=f(g(x))$ for every $x$, then the chain rule is, in Lagrange’s notation,
$$
h^{\prime}(x)=f^{\prime}(g(x)) g^{\prime}(x)
$$
or, equivalently,
$$
h^{\prime}=(f \circ g)^{\prime}=\left(f^{\prime} \circ g\right) \cdot g^{\prime} .
$$
The chain rule may also be expressed in Leibniz’s notation. If a variable $z$ depends on the variable $y$, which itself depends on the variable $x$ (that is, $y$ and $z$ are dependent variables), then $z$ depends on $x$ as well, via the intermediate variable $y$. In this case, the chain rule is expressed as
$$
\frac{d z}{d x}=\frac{d z}{d y} \cdot \frac{d y}{d x}
$$
and
$$
\left.\frac{d z}{d x}\right|{x}=\left.\left.\frac{d z}{d y}\right|{y(x)} \cdot \frac{d y}{d x}\right|_{x}
$$
for indicating at which points the derivatives have to be evaluated.
In integration, the counterpart to the chain rule is the substitution rule.
implicit differentiation
In implicit differentiation, we differentiate each side of an equation with two variables (usually x and y) by treating one of the variables as a function of the other. This calls for using the chain rule.
Find the derivative of the following functions:
a) $\left(x^{2}+2\right)^{2} \quad$ (two methods)
b) $\left(x^{2}+2\right)^{100}$. Which of the two methods from part (a) do you prefer?
Find the derivative of $x^{10}\left(x^{2}+1\right)^{10}$
Find $d y / d x$ for $y=x^{1 / n}$ by implicit differentiation.
Calculate $d y / d x$ for $x^{1 / 3}+y^{1 / 3}=1$ by implicit differentiation. Then solve for $y$ and calculate $y^{\prime}$ using the chain rule. Confirm that your two answers are the same.
Find all points of the curve(s) $\sin x+\sin y=1 / 2$ with horizontal tangent lines. (This is a collection of curves with a periodic, repeated pattern because the equation is unchanged under the transformations $y \rightarrow y+2 \pi$ and $x \rightarrow x+2 \pi$.)
Show that the derivative of an even function is odd and that the derivative of an odd function is even.
(Write the equation that says $f$ is even, and differentiate both sides, using the chain rule.)
1F-7 Evaluate the derivatives. Assume all letters represent constants, except for the independent and dependent variables occurring in the derivative.
a) $D=\sqrt{(x-a)^{2}+y_{0}^{2}}, \quad \frac{d D}{d x}=?$
b) $m=\frac{m_{0}}{\sqrt{1-v^{2} / c^{2}}}, \quad \frac{d m}{d v}=?$
c) $F=\frac{m g}{\left(1+r^{2}\right)^{3 / 2}}, \quad \frac{d F}{d r}=?$
d) $Q=\frac{a t}{\left(1+b t^{2}\right)^{3}}, \quad \frac{d Q}{d t}=?$
