The first principle of a derivative, also known as the definition of a derivative, is the fundamental way to calculate the derivative of a function. It involves finding the limit of the average rate of change of the function as the interval approaches zero. Mathematically, it is expressed as:

$f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}$

Explanation:

• $f'(x)$: This represents the derivative of the function $f(x)$ at the point $x$.
• $h$: This is a small increment in $x$.
• $f(x+h) - f(x)$: This gives the change in the function's value as $x$ changes by $h$.
• $\frac{{f(x+h) - f(x)}}{h}$: This represents the average rate of change of the function over the interval $[x, x+h]$.
• $\lim_{{h \to 0}}$: This limit indicates that $h$ is approaching zero, making the interval infinitesimally small, which gives the instantaneous rate of change, or the derivative, at $x$.

This principle is the basis for all differentiation and is used to determine the slope of the tangent line to the curve at a given point.

Do you want further details or have any questions about this explanation?

Related Questions:

1. How is the first principle of derivatives used to find the derivative of a specific function?
2. What is the geometric interpretation of the derivative using the first principle?
3. How does the first principle differ from using differentiation rules?
4. Can the first principle be applied to non-differentiable functions?
5. How does the concept of limits play a role in the first principle of derivatives?

Tip:

When calculating derivatives using the first principle, it's essential to carefully handle the limit process to ensure accuracy, especially with more complex functions.