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DIFFERENTIABILITY, DIFFERENTIATION RULES AND FORMULAS

Differentiability of a function. A function is said to be differentiable at a point x = x0 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

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