Question

How do you find the derivative of $y = \sqrt x$ using the definition of derivatives?

Answer 1

For solving these types of questions there is no formula in differentiation, therefore we will use the indices formulae of mathematics then we will solve it further by using the differentiation formulae.

Complete step by step Solution:
Given that –
Find the derivative of $y = \sqrt x$
We know that if number written in the square root then the power of that number is the $\dfrac{1}{2}$
Let – a number is the $\sqrt x$ then we can write it is ${x^{\dfrac{1}{2}}}$
Let - $y = \sqrt x$
Now we can write it as $y = {x^{\dfrac{1}{2}}}$
Now we know the formulae of derivation of any variable with the power $n$ is $\dfrac{{dI}}{{dx}}({x^n}) = (n){x^{(n - 1)}}$
Now we will apply above formulae for finding the derivation of $y = {x^{\dfrac{1}{2}}}$ then we will get the derivation of $y$
Now we will derive both side of $y = {x^{\dfrac{1}{2}}}$ with respect to $x$ then we will get
$\Rightarrow \dfrac{{d(y)}}{{dx}} = \dfrac{{d({x^{\dfrac{1}{2}}})}}{{dx}}$
Now we will apply the formula of derivation which is $\dfrac{{dI}}{{dx}}({x^n}) = (n){x^{(n - 1)}}$
Now after derivation both side we will get
$\Rightarrow \dfrac{{d(y)}}{{dx}} = (\dfrac{1}{2}) \times {x^{(\dfrac{1}{2} - 1)}}$
After calculating power, we will get
$\Rightarrow \dfrac{{d(y)}}{{dx}} = (\dfrac{1}{2}) \times {x^{( - \dfrac{1}{2})}}$
After solving the above equation and using the indices formulae of mathematics which we know that ${a^{ - 2}} = \dfrac{1}{{{a^2}}}$ which is power converting formulae in the indices formulae of mathematics we will get
$\Rightarrow \dfrac{{d(y)}}{{dx}} = \dfrac{1}{2}\dfrac{{(1)}}{{{x^{\dfrac{1}{2}}}}}$
Now again we can convert into the form of square root then we will get
$\Rightarrow \dfrac{{d(y)}}{{dx}} = \dfrac{1}{{2\sqrt x }}$

Therefore the derivation of $y = \sqrt x$ is $\dfrac{1}{{2\sqrt x }}$ or we can write it as $\dfrac{1}{{2\sqrt x }}$ which is the required answer of our question

Additional Information:
In these types of questions, we use the basic indices formulae of mathematics which is the most important for the power conversion in mathematics.

Note:
By using law of indices, we know that ${a^{ - 2}} = \dfrac{1}{{{a^2}}}$ which is power converting formulae in the indices formulae of mathematics we will convert given question then we will solve it by using the formulae of derivation which is $\dfrac{{dI}}{{dx}}({x^n}) = (n){x^{(n - 1)}}$ because in differentiation there is no formulae for solving these type question.