Question

Find the derivative of ${{x}^{2}}-2$ at $x=10$?

Answer 1

Hint: For solving these types of problems a good knowledge of differentiation formulae is required and then we can substitute the given value of a variable.
In the above problem, these formulae are used
Derivative of ${{x}^{n}}$ w.r.t x is $\dfrac{d}{dx}\left( {{x}^{n}} \right)=n.{{x}^{n-1}}$& derivative of constant is $\dfrac{d}{dx}\text{(constant)}=0$
Where; $\dfrac{d}{dx}$derivative with respect to x.

Complete step-by-step solution -
Now; let’s solve the problem;
Given expansion is $\mathop{x}^{2}-2$
We have to find the derivative of that expression. So, for our convenience, Let’s consider the total expression equals ‘y’.
$\Rightarrow y=\mathop{x}^{2}-2$
Now, let’s find the derivative of y [i.e ${{x}^{2}}-2$ .] w.r.t x;
$\Rightarrow \dfrac{dy}{dx}=\dfrac{d}{dx}(\mathop{x}^{2}-2)$
$\Rightarrow \dfrac{dy}{dx}=\dfrac{d}{dx}(\mathop{x}^{2})-\dfrac{d}{dx}(2)$
$\Rightarrow \dfrac{dy}{dx}=2\mathop{(x)}^{2-1}-0$ [Using above mentioned formulae]
$\Rightarrow \dfrac{dy}{dx}=2\mathop{x}^{1}-0$
$\Rightarrow \dfrac{dy}{dx}=2x$ …………….…… (1)
In question, they mentioned to find the derivative of expression at $x=10$;
So, assign $x=10$ to variable x in equation 1
$\Rightarrow {{\left( \dfrac{dy}{dx} \right)}_{(x=10)}}=2\times 10$
$\Rightarrow {{\left( \dfrac{dy}{dx} \right)}_{(x=10)}}=20$
Therefore, the derivative of ${{x}^{2}}-2$ at $x=10$ is $''20''$.

Note: Don’t forget to substitute the value of the variable in last because we need to find the derivative at a given value of x.
We can also use the first principle of derivative to find the derivative of the given equation.
A derivative of any function (f(x)) by using first principle formula is
$\Rightarrow f'(x)=\dfrac{f(x+h)-f(x)}{h}$
Where $f'(x)$ is derivative of function (f(x)).
Finally, the above procedure is the best way to solve any derivative problems.
In short, the steps to be followed are:
1. Equate the given expression to a variable ‘y’ (any variable can be taken other than the variable used in an expression).
2. Find the derivative of expression using differentiation formulae and simplify it.
3. If they give any conditions ( like $x=10$ in above problem ); then substitute then in the simplified expression [imp: Don’t substitute the condition before differentiation]