Question

How do I find the derivative of a function at a given point?

Answer 1

Hint: Here we just need to differentiate the given function with respect to the variable which the function contains and then we just need to put the value of the variable at which we need to find the derivative. If we find the differentiation as $f'\left( x \right)$ then at $x = a$ derivative will be $f'\left( a \right)$.

Complete step by step solution:
Here we are given to find the derivative of any function at a given point. Here we must know how the differentiation of any function is carried out. This can be made clearer with an example:
If we have the function $f\left( x \right) = 3{x^2}$ then we know that differentiation of any function of the form ${a^n}{\text{ is }}n{a^{n - 1}}$.
So we can say that $f'\left( x \right) = 6x$ and now if we are given that we need to find the derivative at $x = 2$ then we just need to put the value of $x$ in the derivative of the function which is obtained. So we will get $f'\left( 2 \right) = \left( 6 \right)\left( 2 \right) = 12$.
Another example can be taken as $f\left( x \right) = 6{x^3} + 7x$
Differentiating it with respect with $x$ we get:
$f'\left( x \right) = 18{x^2} + 7$
If we need to find the differentiation or derivative at $x = 1$ then we will get that the derivative will be equal to:
$f'\left( 1 \right) = 18{\left( 1 \right)^2} + 7 = 18 + 7 = 25$.
Hence in this way we just need to find the derivative at any point of the function we are given.

Note:
Here in the similar way we can find the double derivative of any function also. If we are given the derivative as $f'\left( x \right) = 18{x^2} + 7$ then again we can differentiate and get $f''\left( x \right) = 36x$ and now at $x = 1$ derivative will be $f''\left( 1 \right) = 36\left( 1 \right) = 36$.