In calculus, a differentiable function is a one-variable function whose derivative exists at every point in its domain. At each interior point in a differentiable function’s domain, the tangent line to the graph is always non-vertical. There is no break, crest, or angle in a differentiable function. A differentiable function is always continuous; however, not every continuous function is differentiable.

Composite Function- Meaning

The formation of a composite function occurs when a single function is produced by combining two or more functions. Some examples of composite functions are fog(x) or gof(x). 

• The g(x) is the input of the function f(g(x)). f(g(x)) can be also written as fog(x). 
• The f(x) is the input of the function g(f(x)). g(f(x)) can be also written as gof(x). 

Characteristics of Composite Function

The various properties of a composite function include the following:

• Composite functions don’t follow the commutative law. This means that fog ≠ gof. 
• The composite functions follow the associative law, which means (fog)oh = fo(goh). 
• If two functions are bijective, then the composite of both the functions is also bijective. 
• Suppose that there are two functions, f, and g. Both f and g are bijective functions. If gof exists, then (gof)⁻¹ = f⁻¹og⁻¹.
• You will get an even composite function if both f and g are even functions. 
•  You will get an odd composite function if both f and g are odd functions. 
• You will also get an even function if, in case, f is the even function and g is the odd function. 
• You will also get an even function if, in case, f is the odd function and g is the even function. 

Composite Function- Differentiability

A function f(x) is said to be differentiable in its domain if and only if the f(x) exists as the finite derivative. Then f(x) will be differentiable at the point where x = a if the function exists finitely. Hence, the formula of differentiation of the composite functions is,

f'(a) = [f(x) – f(a)]/(x – a)

Differentiable Functions- Properties

The properties of differentiable functions vary according to the different mathematical functions. Some of the important properties of the differentiable functions are as follows:

• You will observe that the sum, product, difference, composite, and quotient of any two differentiable functions is a differential function. 
• A function can be continuous at a point, but the function doesn’t need to be also differentiable at that point. 

Differentiability = continuity

Continuity ≠ differentiability

• A function will not be differentiable if that function is discontinuous at that point. For example, if a function is discontinuous from 3 to 8, that function will not be differentiable from 3 to 8. 

Conclusion

This article explains the differentiability of the composite function. The formation of a composite function occurs when a single function is produced by combining two or more functions. A function f(x) is said to be differentiable in its domain if and only if the f(x) exists as the finite derivative. Then f(x) will be differentiable at the point where x = a if the function exists finitely. The properties of differentiable functions vary according to the different mathematical functions.