Q.Differentiate with respect to $x$ using first principle: $x^{\frac{2}{3}}$.

The key idea is the Definition of the Derivative at a Point (First Principle).

The derivative of a function $f(x)$ with respect to $x$ using the first principle is given by:

For $f(x) = x^{\frac{2}{3}}$, we substitute $f(x)$ and $f(x+h) = (x+h)^{\frac{2}{3}}$ into the definition:

Let $y = x+h$. As $h \to 0$, $y \to x$. The denominator $h$ can be rewritten as $y-x$. The limit becomes:

$f'(x) = \lim_{y \to x} \frac{y^{\frac{2}{3}} - x^{\frac{2}{3}}}{y-x}$ …

The first principle of differentiation is a fundamental concept that defines the derivative of a function. It's not just a formula; it's the very definition of what a derivative is. Geometrically, the derivative of a function $f(x)$ at a point $x$ represents the slope of the tangent line to the curve $y=f(x)$ at that point.

How do we find the slope of a tangent line? We start by approximating it with the slope of a secant line. Consider two points on the curve: $(x, f(x))$ and $(x+h, f(x+h))$. The slope of the secant line connecting these two points is given by:

As the second point $(x+h, f(x+h))$ gets infinitesimally close to the first point $(x, f(x))$, meaning $h$ approaches $0$, the secant line approaches the tangent line. Thus, the slope of the tangent line (the derivative) is the limit of the secant slopes as $h \to 0$.

Let's apply this definition to differentiate $f(x) = x^{2/3}$.

1. Identify the function:

   We are given $f(x) = x^{2/3}$.

2. Determine $f(x+h)$:

   Substitute $x+h$ into the function:

   $f(x+h) = (x+h)^{2/3}$.

3. Set up the limit expression:

   Substitute $f(x+h)$ and $f(x)$ into the first principle formula:

4. Algebraic manipulation to simplify the numerator: This is the crucial step. We need to simplify the expression so that we can evaluate the limit as $h \to 0$ without encountering an indeterminate form like $\frac{0}{0}$. Let $A = (x+h)^{1/3}$ and $B = x^{1/3}$. Then the numerator is $A^2 - B^2$. We can factor this using the difference of squares identity: $a^2 - b^2 = (a-b)(a+b)$. So, the numerator becomes:

Now, substitute this back into the limit expression:

We can rewrite this as a product of two limits (if they both exist):

Let's evaluate the second limit first, as it's straightforward:

Now, we need to deal with the first limit: $\lim_{h \to 0} \frac{(x+h)^{1/3} - x^{1/3}}{h}$.
This is still in the $\frac{0}{0}$ form. To resolve this, we use the difference of cubes identity: $a^3 - b^3 = (a-b)(a^2+ab+b^2)$.
Let $a = (x+h)^{1/3}$ and $b = x^{1/3}$. Then $a-b = (x+h)^{1/3} - x^{1/3}$.
To make the numerator $a^3-b^3$, we multiply the numerator and denominator by $a^2+ab+b^2$:

So, for the first limit term: …