 # Chain rule

A composite function \$y = f(g(x))\$ may be differentiated by using the chain rule:

for example, let \$y = (3x - 2)^3\$.

Let \$g(x) = (3x - 2)\$, and \$f(x) = x^3\$.

Let \$g(x) = u\$, then \$y = u^3\$.

\$\${dy}/{dx} = {dy}/{du}⋅{du}/{dx}\$\$

\${dy}/{du} = 3u^2\$, and \${du}/{dx} = 3\$, so \${dy}/{dx} = {dy}/{du}⋅{du}/{dx} = 3u^2 ⋅ 3 = 9u^2\$

∴ \${dy}/{dx} = 3u^2 = 3(3x - 2)^2\$

The chain rule can also be written as: \$(f o g)'(x) = f'(g(x))⋅ g'(x)\$.

### Derivatives of Odd and Even Functions

The chain rule can be used to demonstrate that the derivative of an odd function is an even function.

If \$f(-x)=-f(x)\$, then the function \$f\$ is odd.

Therefore, \$-f'(x) = f'(x)(-1)\$, and so \$f'(-x)=f'(x)\$, confirming that \$f'\$ is an even function.

And similarly, if \$f\$ is an even function, its derivative is odd.

If \$f\$ is even, \$f(-x)=f(x)\$, and therefore \$f'(-x)(-1)=f'(x)\$. therefore, \$f'(x)=-f'(x)\$, confirming that \$f'\$ is an odd function. Maria Agnesi, 1718 - 1799, was an Italian mathematician, who developed a curve orginally discovered by Fermat and Grandi. The series of functions became known as the witches of Agnesi, since their shapes recall witches' hats, and not because of what she might have had brewing in the kitchen.

The witches of Agnesi are of the form \$y = {a^3}/{x^2 + a^2}\$.

The simplest function, \$y = 1/(x^2+1)\$, can be differentiated by first converting it to a form with rational exponents:

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

Using the chain rule: \$f'(x) = 2x⋅(-1(x^2+1)^{-2}) = -{2x}/{(x^2+1)^{2}}\$

# Product rule

If a function is a product of two functions, the product rule may be used to find the differential.

If \$f(x) = u(x)⋅ v(x)\$, then:

\$\$f'(x) = u(x)v'(x) + u'(x)v(x) \$\$

For example, if \$f(x) = (3x + 2)(2x^2 - 1)\$ then:

\$f'(x) = (3x + 2)4x + 3(2x^2 - 1) \$

\$ = 12x^2 + 8x + 6x'2 - 3 \$

\$ = 18x^2 + 8x -3\$

Alternatively, if \$y = uv\$, then :

\$\${dy}/{dx} = u{dv}/{dx} + v{du}/{dx}\$\$

### Derivation of Product rule

Let \$f(x)=u(x)v(x)\$, where \$u\$ and \$v\$ are differentiable functions.

\$f'(x)={lim}↙{h→0}{u(x+h)v(x+h) -u(x)v(x)}\$

\$={lim}↙{h→0}{u(x+h)v(x+h) -[u(x+h)v(x) -u(x+h)v(x)] -u(x)v(x)}/h\$

\$={lim}↙{h→0}[u(x+h){v(x+h)-v(x)}/h + v(x){u(x+h)-u(x)}/h]\$

\$={lim}↙{h→0}u(x+h)⋅{lim}↙{h→0}{v(x+h)-v(x)}/h \$
\$ + {lim}↙{h→0}v(x)⋅{lim}↙{h→0}{u(x+h)-u(x)}/h\$

\$=u(x)v'(x)+v(x)u'(x)\$

## Second derivative

If \$f(x) = uv\$, then:

\$f'(x) = u'v + uv'\$

Then it follows that:

\$f″(x) = (u'v)' + (uv')' = (u″v+u'v')+(u'v'+uv″) = u″v+2u'v'+uv″\$

The general solution to \$f^n(x)\$ follows the binomial theorem and is called Leibniz's formula.

# Quotient rule

If \$y = u/v\$, \$v(x) ≠ 0\$, then:

\$\${dy}/{dx} = {v(x)u'(x) - v'(x)u(x)}/{(v(x))^2}\$\$

For example, \$y = {x^2 - 1}/{x + 1} = {(x + 1)(x - 1)}/{(x + 1)} = x - 1\$. \$f'(x) = 1\$.

Confirming with the quotient rule: \$u(x) = {x^2 - 1}\$, so \$u'(x) = 2x\$, and \$v(x) = {x + 1}\$, so \$v'(x) = 1\$

\${dy}/{dx} = {v(x)u'(x) - v'(x)u(x)}/{(v(x))^2} = {(x + 1)⋅2x - 1⋅(x^2 - 1)}/{(x + 1)^2} = {(x^2 + 2x + 1)}/{(x + 1)^2} = 1\$

## Implicit functions

An explict function is one where \$y= a_nx^n + a_{n-1}x^{n-1} +.... + a_1x + a_0\$

An implicit function is one such as a circle \$x^2 + y^2 = 1\$, i which y is not explicitly defined in terms of x.

To differentiate an implicit function, differentiate the part with \$y\$ with respect to \$y\$, then multiply this by \${dy}/{dx}\$, according to the chain and product rules.

e.g. \$x^3 + y^3 -2xy^2 = 1\$

\$3x^2 + 3y^2{dy}/{dx} -2(y^2 + 2xy{dy}/{dx}) =0\$

\$3x^2 + 3y^2{dy}/{dx} -2y^2 - 4xy{dy}/{dx} =0\$

\$ (3y^2-4xy){dy}/{dx} =2y^2- 3x^2\$

\${dy}/{dx} = {2y^2- 3x^2}/{3y^2-4xy}\$

## Site Index

### Latest Item on Science Library:

The most recent article is:

Trigonometry

View this item in the topic:

Vectors and Trigonometry

and many more articles in the subject:

### Computing

Information Technology, Computer Science, website design, database management, robotics, new technology, internet and much more. JavaScript, PHP, HTML, CSS, Python, ... Have fun while learning to make your own websites with ScienceLibrary.info. ### Great Scientists

#### Ernst Weizsäcker

1939

Ernst Ulrich von Weizsäcker, b. 1939, is a prominent German scientist and politician. He is a popular author, with best-selling books like 'Factor Four', the update to the Club of Rome 'Limits to Growth', which explains how a sustainable economy requires reductions in consumption and increases in efficiency of this order of magnitude.  ### Quote of the day... You cannot solve a problem using the same sort of thinking that caused the problem in the first place. 