Question

Differentiate xn by initio method

Answer 1

Question:

Differentiate xⁿ by ab initio method.

Solution:

We have to differentiate the given function which is,

$\sf f(x) = x^n$

Differentiation of a function by ab-initio method is given by,

$\boxed{ \tt f'(x) = \lim\limits_{h\to 0} \dfrac{f(x+h) - f(x)}{h} }$

By using this, the derivative of the given function is given by,

${\sf \implies f'(x) =\lim\limits_{h\to0} \dfrac{(x+h)^n - x^n}{h}}$

${\sf \implies f'(x) =\lim\limits_{h\to0} \dfrac{(x+h)^n - (x)^n}{(x + h) - (x)}}$

Since $\tt \red {h \to 0 \implies (x + h) \to x}$

${\sf \implies f'(x) =\lim\limits_{x + h\to x} \dfrac{(x+h)^n - (x)^n}{(x + h) - (x)}}$

Now we can apply standard limit,

$\boxed{\tt\lim_{x\to a} \dfrac{x^n - a^n}{x - a} = n \cdot a^{n-1}}$

Using this, we get:

${\sf \implies f'(x) =n \cdot x^{n - 1} }$

Hence the derivative of $\sf x^n$ is $\sf n \cdot x^{n-1}$.