Question

How do you find the derivative of $\dfrac{4}{{\sqrt x }}$?

Answer 1

Hint: We are given an expression and we have to find its derivative. When we are differentiating things involving radicals it's never a bad idea to rewrite the radicals as rational exponents. After it becomes easy to differentiate using the power rule i.e.
$\dfrac{d}{{dx}}a{x^n} = an{x^{n - 1}}$
To use the power rule, multiply the variable’s exponent n, by its coefficient a, then subtract 1 from the exponent. If there is no coefficient (the coefficient is 1), then the exponent will become the new coefficient.
Here $x$ is any variable and $n$ is the exponent. By using this we will find the derivative of the expression.

Complete step-by-step answer:
Step 1: We are given a mathematical expression i.e. $\dfrac{4}{{\sqrt x }}$ and we have to find its derivative. Firstly we will rewrite the expression in the radical form i.e. rational number having an exponent.
$\dfrac{4}{{\sqrt x }} = 4x^{-1/2}$
$\Rightarrow y= 4x^{-1/2}$
Step 2: Now, this is easily differentiated using the power rule:
$\dfrac{d}{{dx}}a{x^n} = an{x^{n - 1}}$
Here, $a = 4;n = - \dfrac{1}{2};x = x$
Using this we will differentiate:
$\dfrac{dy}{dx}= \dfrac{-1}{2}.4x^{-3/2}$
Now we will simplify it:
$\dfrac{dy}{dx}= -2x^{-3/2}$
And we can convert it in this form also
$\dfrac{{dy}}{{dx}} = - \dfrac{2}{{\sqrt {{x^3}} }}$

Hence the derivative is $ - \dfrac{2}{{\sqrt {{x^3}} }}$

Note:
This type of question has an expression which can be converted into radical form then just use a power rule and solve it. There is no easier way than this sometime students try to apply first principles to find the derivative but unless it's mentioned in the question to solve by first principle don't use it because it makes a question lengthy and sometimes so messed up. Students mainly make mistakes only. In these questions, use a simple power rule which is short and easy.
Commit to memory:
$\dfrac{d}{{dx}}a{x^n} = an{x^{n - 1}}$