Derivative of x^3/2: by First Principle, Power Rule

The derivative of x^3/2 is equal to $\frac{3}{2} x^{1/2}$. In this blog post, we will find the derivative of x to the power 3/2 using the first principle and power rule.

To find the derivative of $x^{\frac{3}{2}}$ using the limit definition, let us first recall the first principle of derivatives. The rule says that the derivative of a function $f(x)$ by the first principle is given by

$\dfrac{d}{dx}(f(x))$ $=\lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$ $\cdots (I)$

Derivative of x^3/2 by First Principle

We will follow the below steps to find the derivative of $x^{\frac{3}{2}}$ by the first principle.

Step 1: Let us put $f(x)=x^{\frac{3}{2}}$ in (I). Thus, the derivative of $x^{\frac{3}{2}}$ using the first principle will be given as follows.

$\dfrac{d}{dx}(x^{\frac{3}{2}})$ $=\lim\limits_{h \to 0} \dfrac{(x+h)^{\frac{3}{2}}-x^{\frac{3}{2}}}{h}$

Step 2: Multiplying both the numera\tor and the denomina\tor by $((x+h)^{\frac{3}{2}}+x^{\frac{3}{2}})$, the derivative is

$\dfrac{d}{dx}(x^{\frac{3}{2}})$ $=\lim\limits_{h \to 0} [ \dfrac{(x+h)^{\frac{3}{2}}-x^{\frac{3}{2}}}{h}$ $\times \dfrac{(x+h)^{\frac{3}{2}}+x^{\frac{3}{2}}}{(x+h)^{\frac{3}{2}}+x^{\frac{3}{2}}}]$

$=\lim\limits_{h \to 0} \dfrac{[(x+h)^{\frac{3}{2}}]^2-[x^{\frac{3}{2}}]^2}{h[(x+h)^{\frac{3}{2}}+x^{\frac{3}{2}}]}$ by the formula $(a-b)(a+b)$ $=a^2-b^2$.

Step 3: Applying the rule of indices $(a^m)^n =a^{mn}$, we obatin that

$\dfrac{d}{dx}(x^{\frac{3}{2}})$ $=\lim\limits_{h \to 0} \dfrac{(x+h)^3-x^3}{h[(x+h)^{\frac{3}{2}}+x^{\frac{3}{2}}]}$

$=\lim\limits_{h \to 0} \dfrac{x^3+3x^2h+3xh^2+h^3-x^3}{h[(x+h)^{\frac{3}{2}}+x^{\frac{3}{2}}]}$ using the formula $(a+b)^3$ $a^3+3a^2b+3ab^2+b^3$.

$=\lim\limits_{h \to 0} \dfrac{3x^2h+3xh^2+h^3}{h[(x+h)^{\frac{3}{2}}+x^{\frac{3}{2}}]}$

$=\lim\limits_{h \to 0} \dfrac{h(3x^2+3xh+h^2)}{h[(x+h)^{\frac{3}{2}}+x^{\frac{3}{2}}]}$

$=\lim\limits_{h \to 0} \dfrac{3x^2+3xh+h^2}{(x+h)^{\frac{3}{2}}+x^{\frac{3}{2}}}$

Step 4: We now substitute $h=0$ to get the limit. By doing so, we obtain that

$\dfrac{d}{dx}(x^{\frac{3}{2}})$ $=\dfrac{3x^2+3x\cdot 0+0^2}{(x+0)^{\frac{3}{2}}+x^{\frac{3}{2}}}$

$=\dfrac{3x^2}{2x^{\frac{3}{2}}}$

$=\frac{3}{2} x^{2-\frac{3}{2}}$

$=\frac{3}{2} x^{1/2}=\frac{3}{2} \sqrt{x}$

Thus, the derivative of $x^{\frac{3}{2}}$ by the first principle is equal to $\frac{3}{2} \sqrt{x}$.

Derivative of x^3/2 by Power Rule

The power rule of derivatives says that the derivative of $x^n$ is given by the formula:

$\dfrac{d}{dx}(x^n)=nx^{n-1}$.

Putting n=3/2 in this formula, we will get the derivative of x^3/2. Thus,

$\dfrac{d}{dx}(x^{\frac{3}{2}})$ $=\frac{3}{2} x^{\frac{3}{2}-1}$

$=\frac{3}{2} x^{1/2}$

$=\frac{3}{2} \sqrt{x}$.

So the derivative of $x^{\frac{3}{2}}$ by power rule is equal to $\frac{3}{2} \sqrt{x}$.

Also Read: 

Derivative of xe^x

Derivative of 10^x

Derivative of 1/(1+x²)

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