Question

How do you calculate derivatives?

Answer 1

Hint: We first explain the form of derivative with respect to the slope. We express the mathematical form with both $\dfrac{dy}{dx}$ and $slope=\dfrac{\Delta y}{\Delta x}$. We then represent the small changes for the limit theorem of derivative. We understand the concept better with an example.

Complete step-by-step solution:
Let us take a function in the form of $y=f\left( x \right)$. We try to find the slope of the curve at a particular point of $\left( a,b \right)$ which is on the curve.
The slope will be considered as the unit change of $y$ with respect to change of $x$.
The mathematical representation will be done for small changes where the slope becomes $\dfrac{dy}{dx}$.
We can also express it as $slope=\dfrac{\Delta y}{\Delta x}$. The derivative of a function is the rate at which the function value is changing, with respect to x, at a given value of x.
For the point $\left( a,b \right)$, the slope will be \[{{\left[ \dfrac{dy}{dx} \right]}_{\left( a,b \right)}}\].
This representation of $\dfrac{dy}{dx}$ is called the derivative form.
Let us take an example where we take $f\left( x \right)=\log x$. The slope or derivative form of the function gives $\dfrac{dy}{dx}={{f}^{'}}\left( x \right)=\dfrac{1}{x}$.
For the point $\left( 1,1 \right)$, the slope will be \[{{\left[ \dfrac{dy}{dx} \right]}_{\left( 1,1 \right)}}={{\left[ \dfrac{1}{x} \right]}_{\left( 1,1 \right)}}=1\].

Note: The derivative has a limit definition for $slope=\dfrac{\Delta y}{\Delta x}$. The function changes from ${{y}_{1}}=f\left( x \right)$ to ${{y}_{2}}=f\left( x+h \right)$ where $h\to 0$.
So, $slope=\dfrac{\Delta y}{\Delta x}=\displaystyle \lim_{h \to 0}\dfrac{f\left( x+h \right)-f\left( x \right)}{h}$.