Question

Does differentiate mean find the derivative?

Answer 1

Hint: Sometimes this question arises in our mind because different books use different terms that's why we are confused whether to call it differentiate or derivative, if they both are the same or different.

Complete step by step solution:
Yes, differentiate means finding the derivative of a function. In mathematics, actually finding the derivative of a function is nothing but known as differentiation of function. The essence of calculus is the derivative. The derivative is the instantaneous rate of change to one of its variables. This is equivalent to finding the slope of the tangent line to the function at a point. As we describe above, function represents the rate of change of the function, which is nothing, but the differentiation of function.
The differentiation and derivative of the function is denotes as
$ \Rightarrow \dfrac{d\left( f\left( x \right) \right)}{dx} $ or $ f' \left( x \right) $
Let's take a small example to find the derivative or differentiation.

Example- Find the derivative or differentiation of $ y=3{{x}^{2}} $ ?

Solution- Here we are seeing $ y $ is a dependent variable and $ x $ is the independent variable, it means as we increase the value of $ x $ , $ y $ also increases. Similarly if $ x $ decreases $ y $ also decreases.
Since $ 3 $ is constant with respect to $ x $ , the derivative or differentiation of $ 3{{x}^{2}} $ with respect to $ x $ is:
$ \Rightarrow \dfrac{d\left( 3{{x}^{2}} \right)}{dx}=3\dfrac{d\left( {{x}^{2}} \right)}{dx} $
Now we very well know that differentiation or derivative of $ {{x}^{2}} $ is $ 2x $ , so by putting this in above equation, we get
$ \Rightarrow \dfrac{dy}{dx}=3\cdot 2x=6x $
Hence the derivative or differentiation of $ y=3{{x}^{2}} $ is $ 6x $.

Note: Here we also have to know that in calculus differentiation is finding the slope of the function. And the derivative represents how fast something is changing at an instant. We should know the derivative of position is speed and derivative of speed is acceleration.