SolitaryRoad.com

Website owner: James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

DIFFERENTIABILITY, DIFFERENTIATION RULES AND FORMULAS

Differentiability of a function. A function is said to be differentiable at a point x =
x_{0} if it has a derivative there. A function is said to be differentiable on an interval if it is
differentiable at every point of the interval.

In general a function is differentiable on an interval if the function is smooth on that interval, meaning it has no abrupt changes in direction on the interval. If a function has a sharp change in direction at some point the derivative won’t exist at that point. The functions encountered in elementary calculus are in general differentiable, except possibly at certain isolated points on their intervals of definition.

Differentiation rules and formulas. In the following rules and formulas u and v are differentiable functions of x while a and c are constants.

The derivative of a constant is zero.

The derivative of a variable with respect to itself is one.

The derivative of the sum of two functions is equal to the sum of their separate derivatives.

The derivative of the product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.

The derivative of the product of a constant and a function is the constant times the derivative of the function.

The derivative of the quotient of a function by a constant is the derivative of the function divided by the constant.

The derivative of y with respect to x is the reciprocal of the derivative of x with respect to y.

The derivative of the quotient of two functions is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

9. Let y be a function of u and u be a function of x i.e. y = f(u) and u = g(x). Then

where n is any real number (positive, negative, fractional, etc).

The derivative of the n-th power of x is n times the (n-1)th power of x.

where n is any real number (positive, negative, fractional, etc).

The derivative of the n-th power of a function is n times the (n-1)th power of the function times the derivative of the function.

Trigonometric functions

Inverse trigonometric functions

The inverse trigonometric functions are multi-valued. The principal branches are as follows.

Exponential and logarithmic functions

where a is a constant.

Hyperbolic functions

Inverse hyperbolic functions

More from SolitaryRoad.com:

Jesus Christ and His Teachings

Way of enlightenment, wisdom, and understanding

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

On Self-sufficient Country Living, Homesteading

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

Theory on the Formation of Character

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-control, self-restraint, self-discipline basic to so much in life

[ Home ] [ Up ] [ Info ] [ Mail ]