Derivative of x root(x): by First Principle, Power Rule

The derivative x root(x) is equal to (3√x)/2. Note x root(x) is denoted by x√x, and its derivative as d/dx (x√x). In this post, let us learn how to differentiate x root(x) using the following methods:

1. By power rule
2. By first principle.

The derivative formula of x root(x) is given as follows.

$\dfrac{d}{dx}(x\sqrt{x}) = \dfrac{3\sqrt{x}}{2}.$

By Power Rule

To find the derivative of x√x, let us first write it as a power of x. Note that √x = x^1/2. So we obtain that

x√x = x ⋅ x^1/2 = x^{1+ 1/2} = x^3/2.

Differentiating both sides, we get that

$\dfrac{d}{dx}(x\sqrt{x}) = \dfrac{d}{dx}(x^{3/2})$ = $\dfrac{3}{2} x^{\frac{3}{2}-1}$ by the power rule d/dx (xn) = nxn-1. = $\dfrac{3}{2} x^{\frac{1}{2}}$ = $\dfrac{3\sqrt{x}}{2}$.

So the derivative of x root(x) by the power rule is (3√x)/2.

Also Read: Derivative of fourth root of x

Derivative of x root(x) by First Principle

The derivative of f(x) by first principle is denoted by d/dx (f(x)) and it is given by the limit formula below:

$\dfrac{d}{dx} (f(x))$ = lim_h→0 $\dfrac{f(x+h)-f(x)}{h}$.

Let f(x) = x√x. Note that

f(x) = x^3/2 and f(x+h) = (x+h)^3/2.

Then by first principle, the differentiation of x√x is

$\dfrac{d}{dx}(x\sqrt{x})$ = limh→0 $\dfrac{(x+h)^{\frac{3}{2}}-x^\frac{3}{2}}{h}$

Put x+h = z. So z→x as h→0. Also, h=z-x.

Now, using the limit formula lim_x→a (xⁿ-aⁿ)/(x-a) = na^n-1, we deduce from above that

$\dfrac{d}{dx}(x\sqrt{x})$ = limh→0 $\dfrac{z^{\frac{3}{2}}-x^\frac{3}{2}}{z-x}$ = $\dfrac{3}{2} x^{\frac{3}{2}-1}$ = $\dfrac{3\sqrt{x}}{2}$.

Therefore, d/dx (x√x) = (3√x)/2.

So the derivative of x root(x) by first principle is (3√x)/2.

Related Topics:

Derivative of sin3x : The derivative of sin3x is 3cos3x.

Derivative of e^sinx : The derivative of e^sinx is cosx e^sinx.

By Logarithmic Differentiation

Let y = x√x = x^3/2

Taking natural logarithms on both sides, we get that

ln(y) = ln(x^3/2)

⇒ ln(y) = $\dfrac{3}{2}$ ln(x) by the log rule lnx^k = k lnx.

Differentiating both sides with respect to x, we get that

$\dfrac{1}{y} \dfrac{dy}{dx} = \dfrac{3}{2} \cdot \dfrac{1}{x}$

⇒ $\dfrac{dy}{dx} = \dfrac{3y}{2x}= \dfrac{3x^{3/2}}{2x} = \dfrac{3\sqrt{x}}{2}$.

Hence, the derivative of x root(x) by the logarithmic differentiation is equal to (3√x)/2.

More Reading:

Derivative of 1/x : The derivative of 1/x is -1/x².

Derivative of ln5x : The derivative of ln(5x) is 1/x.

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