Explanation on Differentiability

A differentiable function of one real variable is one that has a derivative at each point in its domain. In other words, a differentiable function’s graph contains a non-vertical tangent line at each interior point in its domain. A differentiable function has no break, angle, or cusp and is smooth (it can be approximated locally as a linear function at each interior point). Differentiation is extremely important, and being able to determine whether a given function is differentiable is a crucial skill. Because of this, the derivative of a function is frequently referred to as the “instantaneous rate of change.”It calculates the rate of change of the function value in relation to its input.”

Differentiability

When there is a defined derivative at a point in a function, that point is differentiable. This means that the slope of the tangent line connecting the points on the left is approaching the same value as the slope of the tangent line connecting the points on the right.

Differentiable Functions Rules

If f and g are differentiable functions, we can apply some rules to find the derivatives of their sum, difference, product, and quotient. Here are some differentiability formulas for determining a differentiable function’s derivatives:

• (f + g)’ = f’ + g’

• (f – g)’ = f’ – g’

• (fg)’ = f’g + fg’

• (f/g)’ = (f’g – fg’)/f2

Differentiability formula

Differentiation of differentiable functions is a mathematical process that determines the rate of change of the functions with respect to a variable in calculus. The following are some common differentiability formulas that we use to solve various mathematical problems:

The following is the derivation of 

• sin x: (sin x)’ = cos x

• Cos x derivative: (cos x)’ = -sin x

• tan x derivative: (tan x)’ = sec2 x

• Cot x derivative: (cot x)’ = -cosec2 x

• Sec x derivation: (sec x)

• Cosec x derivative: (cosec x)’ = -cosec x.cot x

• xn derivative: (xn)’ = nxn-1

• (ex)’ = ex’s derivative: ex’s derivative

• ln x derivative: (ln x)’ = 1/x

Differentiability of a function

A differentiable function of one real variable is one that has a derivative at each point in its domain. In other words, a differentiable function’s graph contains a non-vertical tangent line at each interior point in its domain.

If the derivative of a function exists at all points in its domain, the function is said to be differentiable. If a function f(x) can be differentiated at x = a, then f′(a) exists in the domain. Consider the following examples of differentiable polynomial and transcendental functions:

• f(x) = x4 – 3x + 5

• f(x) = x100

• f(x) = sin x

• f(x) = ex

Differentiable Functions Limit Formula

The limits can also be used to determine whether a function f(x) is differentiable. At the point x = a, a function f(x) is differentiable if and only if the following limit exists:

lim h → 0 f ( c + h ) −f ( c ) h /h

Differentiable Functions Tricks

• If a graph has a sharp corner at a point, the function at that point is not differentiable.

• If a graph has a break at a point, the function at that point is not differentiable.

• If a point on a graph has a vertical tangent line, the function is not differentiable at that point.

Differentiability Importance

A function’s differentiability is important in many analyses because many theorems (such as Rolle’s Theorem) simply do not hold when a function is not differentiable. It’s essentially so that we can assume the function is “well-behaved” in such a way that we can apply standard theorems to it; otherwise, some of these theorems are no longer applicable. Also, continuous curves can have strange properties that we may want to avoid; for example, we may not want functions to have sharp (but still continuous) corners.

Conclusion

A differentiable function of one real variable has a derivative at every point in its domain. In other words, a differentiable function’s graph contains a non-vertical tangent line at each interior point in its domain. A differentiable function has no break, angle, or cusp and is smooth (it can be approximated locally as a linear function at each interior point). Differentiation is critical, and knowing whether or not a given function is differentiable is a critical skill. 

Differentiation is extremely important, and being able to determine whether a given function is differentiable is a crucial skill. Because of this, the derivative of a function is frequently referred to as the “immediate rate of change.” It calculates the rate at which the function value changes in relation to its input.