Differentiable

Differentiable
Written by: Jack Methew

Jack Methew knows that successful students become successful adults. This is her 15th year at Edison Elementary School and her 10th year teaching fourth grade. So far, fourth grade is her favorite grade to teach! Mrs. Carroll was the 2011 Newell Unified School District Teacher of the Year, and received her National Board Certification in 2013. She loves science and majored in biology at Arizona State University, where she also earned her teaching credential and Master of Education degree. Mrs. Carroll is excited to begin the best year ever!

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Definition of Differentiable:

Differentiability is in context to a function, we will always be considering a function and then will be telling about its differentiability.

So let's see the definition of differentiable

“Let f(x) be a real valued function defined on (a,b) and let c∈(a,b) then f(x) is said to be derivable or differentiable function at x = c, iff following exists finitly:”

$$ \lim\limits_{x \to c} \frac{f(x)-f(c)}{x-c} $$

We have a function given to us and that is a real valued function f(x)and it is defined on the open interval (a,b). Now, there exists a point "c" on that interval of (a,b) i.e. c∈(a,b), then the function will be differentiable if it exists on the above limit for finite quantity.

Notation of Differentiable Function:

Now how to represent differentiability if I have a function f(x) or f(c) then its notation will be:

$$ f'(x) = f'(c) = Df(c) = \left| \frac{df(x)}{dx} \right|_{x=c} $$

Condition for Differentiability:

We talk earlier that if limit exists, what limit exists means the left hand limit should be equal to the right hand limit and in case of differentiability we say that the left hand derivative should be equal to the right hand derivative then we say that the limit exists finitely means it should be a finite quantity. So, the main idea is to find the derivative of a function.

Mathematically,

$$ \lim\limits_{x \to c^-} \frac{f(x)-f(c)}{x-c} = \lim\limits_{x \to c^+} \frac{f(x)-f(c)}{x-c} $$ $$ \lim\limits_{h \to 0} \frac{f(c-h)-f(c)}{c-h-c} = \lim\limits_{h \to 0} \frac{f(c+h)-f(c)}{c+h-c} $$ $$ \lim\limits_{h \to 0} \frac{f(c-h)-f(c)}{-h} = \lim\limits_{h \to 0} \frac{f(c+h)-f(c)}{h} $$

Let’s try to deep understanding by some practical meaning

Differentiable Function Example:

Consider we have a function:

$$ f(x) = \begin{cases} x, & x \le 1 \\ x^3, & x \gt 1 \end{cases} $$

Now for testing differentiability, Let's find the derivative of a function

$$ f'(x) = \begin{cases} 1, & x \le 1 \\ 3x^2, & x \gt 1 \end{cases} $$

Now, let’s see what is the left hand derivative and also right hand derivative

The left-hand derivative of a function is:

$$ \lim\limits_{x \to 1^-} f'(x)=1 $$

Also, The right-hand derivative of a function is:

$$ \lim\limits_{x \to 1^+} f'(x)=3(1)^2 $$ $$ \lim\limits_{x \to 1^+} f'(x)=3 $$

So, both side limits are not the same which shows that the function is not a differentiable function.