Properties of Differentiability

Differentiability describes whether a function has a well-defined derivative at a point or over an interval. Differentiability is a property of a function that indicates how smooth a function is. If a function is differentiable at a point or an interval, it means that you can find its derivative.  A function f is differentiable at a point c if:

• The function is smooth at c with no sharp corners, discontinuities or cusps
• Derivative f'(c) exists at that point, which means that the limit that defines the derivative
• If the function is differentiated at every point in its domain. 

$f^{\text{'}} \left( c \right) = \underset{ h \to 0 }{lim} \frac{ f ( c + h ) - f \left( c \right) }{h}$

Many important theorems are associated with differentiability to explain the behaviour of differentiable functions. The CBSE board ask questions based on the differentiability theorem. 

Let us take a look at these:

1. Sum Rule

In case f and g are differentiable at x, in that case, the sum f + g will also be differentiable at x. Here, the derivative of the sum is the sum of derivatives (f+g)′(x)=f′(x)+g′(x). Through this theorem, it is possible to differentiate functions which are sums of other differentiable functions.

2. Product Rule

If f and g are differentiable at x, then the product f.g is also differentiable at x. The derivative of product is given by (fg)′(x)=f′(x)g(x)+f(x)g′(x). This rule is important for differentiating functions that are products of other differentiable functions.

3. Quotient Rule

In case f and g are differentiable at x and g(x) ≠ 0. Then, the quotient f/g will also be differentiable at x and the derivative of quotient is given by 

${ \left( \frac{f}{g} \right) }^{\prime } \left(x\right) =$ = $\frac{ f^{\prime } \left(x\right) g\left(x\right) - f\left(x\right) g^{\prime } \left(x\right) }{ { \left[ g\right(x\left) \right] }^2 }$

Through this theorem, it is possible to differentiate functions that are ratios of other differentiable functions.

4. Chain Rule

If f is differentiable at x whereas g is differentiable at f(x), then the composition g∘f will be differentiable at x. In this case, the derivative of the composition is given by (g∘f)′(x)=g′(f(x))⋅f′(x).  This rule is important for differentiating composite functions, which allow us to break down complex functions into simpler parts. 

If it is said that a function is differentiable over an interval, it indicates that a function has a well-defined derivative at every point within that interval. The NEET exam or JEE Main exam will ask questions around this topic.

 function will be differentiable on an interval (a,b) if it: 

• is differentiable at every point c in (a,b) 
• includes its endpoints [a,b]. We will, then, consider, one-sided derivatives at a and b:
• at x = a, the right-hand derivative must exist
• at x = b, left-hand deriavtive must exist

Differentiability means there will be continuity. If f is differentiable on [a,b]; it will also be continuous on [a,b].

Suppose, f is differentiable on (a,b) and it is continuous on [a,b], then Mean Value Theorem guarantees the existence of point c in (a,b) where $f^{\text{'}} \left( c \right) = \underset{ h \to 0 }{lim} \frac{ f ( c + h ) - f \left( c \right) }{h}$

Let us understand the importance of differentiability:

• If f is differentiable at c, the graph of f will have a well-defined tangent line at (c,f(c)). Slope of this tangent line is f'(c)
• Differentiability ensures that quantities such as velocity and acceleration exist. 
• Through differentiability, it is possible to find critical points where f'(x) = 0 or f'(x) is undefined. 
• Differentiable functions can be approximated by polynomials that are useful for numerical analysis.
• Differentiability also ensures that a function is smooth enough to have well-defined rate of change, which is important for modelling real-world phenomena where predictability is required.

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