Prove That H is Defined and Continuous on All of R Prove That H is Differnetiable on R

Differentiable

A differentiable function is a function in one variable in calculus such that its derivative exists at each point in its entire domain. The tangent line to the graph of a differentiable function is always non-vertical at each interior point in its domain. A differentiable function does not have any break, cusp, or angle. A differentiable function is always continuous but every continuous function is not differentiable.

In this article, we will explore the meaning of differentiable, how to use differentiability rules to find if the function is differentiable, understand the importance of limits in differentiability, and discover other interesting aspects of it.

1.	What is Differentiable?
2.	Rules for Differentiable Functions
3.	Some Common Differentiability Formulas
4.	Limit Formula for Differentiable Functions
5.	Difference Between Differentiable and Continuous Function
6.	FAQs on Differentiable

What is Differentiable?

A function is said to be differentiable if the derivative of the function exists at all points in its domain. Particularly, if a function f(x) is differentiable at x = a, then f′(a) exists in the domain. Let us look at some examples of polynomial and transcendental functions that are differentiable:

f(x) = x⁴ - 3x + 5
f(x) = x¹⁰⁰
f(x) = sin x
f(x) = e^x

Rules for Differentiable Functions

If f, g are differentiable functions, then we can use some rules to determine the derivatives of their sum, difference, product and quotient. Here are some differentiability formulas used to find the derivatives of a differentiable function:

(f + g)' = f' + g'
(f - g)' = f' - g'
(fg)' = f'g + fg'
(f/g)' = (f'g - fg')/f²

Example

Let's use the differentiability rules to find the derivative of the function f(x) = (2x+1)³

df/dx = d(2x+1)³/dx

= d(8x³ + 12x² + 6x + 1)/dx

= 24x²+ 24x + 6

= 6(2x+1)²

Some Common Differentiability Formulas

In calculus, differentiation of differentiable functions is a mathematical process of determining the rate of change of the functions with respect to the variable. Some common differentiability formulas that we use to solve various mathematical problems are:

Derivation of sin x: (sin x)' = cos x
Derivative of cos x: (cos x)' = -sin x
Derivative of tan x: (tan x)' = sec² x
Derivative of cot x: (cot x)' = -cosec² x
Derivative of sec x: (sec x)' = sec x.tan x
Derivative of cosec x: (cosec x)' = -cosec x.cot x
Derivative of xⁿ: (xⁿ)' = nx^n-1
Derivative of e^x: (e^x)' = e^x
Derivative of ln x: (ln x)' = 1/x

Limit Formula for Differentiable Functions

There is an alternative way to determine if a function f(x) is differentiable using the limits. A function f(x) is differentiable at the point x = a if the following limit exists:

\[\lim_{h\rightarrow 0}\dfrac{f(c+h)-f(c)}{h}\]

Example: Consider the absolute value function given by f(x) = |x|

We will determine if this function is differentiable at c = 0 or not. Let's find the limit \(\begin{align}\lim_{h\rightarrow 0}\dfrac{f(c+h)-f(c)}{h}\end{align}\).

\[\begin{align}\lim_{h\rightarrow 0}\dfrac{f(c+h)-f(c)}{h}&=\lim_{h\rightarrow 0}\dfrac{f(0+h)-f(0)}{h}\\&=\lim_{h\rightarrow 0}\dfrac{|h|-|0|}{h}\\&=\lim_{h\rightarrow 0}\dfrac{|h|}{h}\end{align}\]

What happens when \(h\) approaches 0 from left? Let's see the behavior of the function as h becomes closer to 0 from the negative x - axis.

\[\begin{align}\lim_{h\rightarrow 0^{-}}\dfrac{|h|}{h}&=\lim_{h\rightarrow 0}\dfrac{-h}{h}\\&=-1\end{align}\]

What happens when h approaches 0 from right? Now, let's see the behavior of the function as h becomes closer to 0 from the positive x - axis.

\[\begin{align}\lim_{h\rightarrow 0^{+}}\dfrac{|h|}{h}&=\lim_{h\rightarrow 0}\dfrac{h}{h}\\&=1\end{align}\]

Did you observe that limits are different?

This means that the limit \(\begin{align}\lim_{h\rightarrow 0}\dfrac{f(c+h)-f(c)}{h}\end{align}\) does not exists at c = 0 for f(x) = |x|.

This implies that the absolute value function f(x) = |x| is not differentiable at x = 0.

Difference Between Differentiable and Continuous Function

We say that a function is continuous at a point if its graph is unbroken at that point. A differentiable function is always a continuous function but a continuous function is not necessarily differentiable.

Example

We already discussed the differentiability of the absolute value function. Clearly, there are no breaks in the graph of the absolute value function. The function is continuous everywhere. Particularly, the function is continuous at x=0 but not differentiable at x=0.

Hence the main difference between a differentiable and continuous function is that a differentiable function is always a continuous function but a continuous function may not be differentiable.

Tips and Tricks for Differentiable Functions

If a graph has a sharp corner at a point, then the function is not differentiable at that point.
If a graph has a break at a point, then the function is not differentiable at that point.
If a graph has a vertical tangent line at a point, then the function is not differentiable at that point.

Important Notes on Differentiable

Differentiable functions are those functions whose derivatives exist.
If a function is differentiable, then it is continuous.
If a function is continuous, then it is not necessarily differentiable.
The graph of a differentiable function does not have breaks, corners, or cusps.

Related Topics on Differentiable

Differentiation of Trigonometric Functions
Derivative of ln x
Derivative Formula
Differentiation

Differentiable Examples

Example 1: Use the differentiability rules to determine the derivative of f(x) = (2x + 1)/x³

Solution: We will use the quotient rule for differentiable functions to determine the derivative of f(x).

df/dx = d((2x + 1)/x³)/dx

= [2x³ - 3x².(2x + 1)]/x⁶

= [2x³ - 6x³- 3x²]/x⁶

= -(4x³ + 3x²)/x⁶

Answer: The derivative of f(x) = (2x + 1)/x³is -(4x³ + 3x²)/x⁶
Example 2: Find out where the given function f(x) = |x + 2| is not differentiable using graph and limit definition.

Solution:

Clearly, there is a sharp corner at point x = -2. The function is not differentiable at x = -2. Now, let's use the limit definition of differentiable functions. We already observed that the limits are different for absolute value of function. \[\begin{align}\lim_{h\rightarrow 0}\dfrac{f(c+h)-f(c)}{h}&=\lim_{h\rightarrow 0}\dfrac{f(-2+h)-f(-2)}{h}\\&=\lim_{h\rightarrow 0}\dfrac{|h|-0}{h}\\&=\lim_{h\rightarrow 0}\dfrac{|h|}{h}\end{align}\]

This means that the limit \(\begin{align}\lim_{h\rightarrow 0}\dfrac{f(c+h)-f(c)}{h}\end{align}\) does not exists at c = -2 for f(x) = |x+2|.

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Differentiable Practice Questions

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FAQs on Differentiable

What is Differentiable in Calculus?

A function is said to be differentiable if the derivative of the function exists at all points in its domain.

Is the Cubic Function Differentiable?

Yes, the cubic function is differentiable. For example, the function f(x) = x³ is differentiable and its derivative is f′(x) = 3x²

What does Twice Differentiable Mean?

If a function is twice differentiable, then it means that the second derivative of the function exists.

Why is the Absolute Value Function not Differentiable at 0?

The absolute value function is not differentiable at 0 because the graph of the function has a sharp corner at this point.

What are Some Common Differentiable Formulas?

Derivation of sin x: (sin x)' = cos x
Derivative of cos x: (cos x)' = -sin x
Derivative of tan x: (tan x)' = sec² x
Derivative of cot x: (cot x)' = -cosec² x
Derivative of sec x: (sec x)' = sec x.tan x
Derivative of cosec x: (cosec x)' = -cosec x.cot x
Derivative of xⁿ: (xⁿ)' = nx^n-1
Derivative of e^x: (e^x)' = e^x
Derivative of ln x: (ln x)' = 1/x

When is a Function Differentiable?

A differentiable function is a function in one variable in calculus such that its derivative exists at each point in its entire domain.

How to Prove a Function is Differentiable?

A function can be proved differentiable if its left-hand limit is equal to the right-hand limit and the derivative exists at each interior point of the domain.

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Source: https://www.cuemath.com/calculus/differentiable/