## 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) $Q=\frac{a t}{\left(1+b t^{2}\right)^{3}}, \quad \frac{d Q}{d t}=?$

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