What is differentiation?

Differentiation is a mathematical concept in Calculus that involves finding the rate at which a function changes with respect to its input. It is also commonly known as the derivative and plays a crucial role in various practical applications. Derivatives help to calculate the slope or gradient of a curve at a specific point, representing the function's instantaneous rate of change at that point.

Differentiation as limits

We will first try to demonstrate how to find the rate of change at a specific point first graphically. In simple terms, it is tangent to the curve where we are trying to find the rate of change with respect to $x$.

In the diagram above, we aim to determine the derivative at point $x$. To precisely calculate the rate of change at that particular point, we must decrease the value of $\Delta x$ so that it becomes very small as if we are finding the rate of change at that exact point.

In formal terms, we can represent a function's derivative as $f'(x)$ which is called the first derivative of the function.

As the value of $\Delta x$ gets closer to zero, we can determine the derivative at specific points along a curve. A function is called differentiable if both the left-hand limit and the right-hand limit as $\Delta x$ approaches zero are equal.

Differentiation rules

Calculating all derivatives of a function using the formal definition can be somewhat difficult, although it is the base of differentiation. However, several rules allow us to find derivatives directly. The image below illustrates these rules, which are essential for certain types of functions, enabling us to calculate derivatives more efficiently and easily.

Power rule

The power rule of differentiation states that the derivative of a power function, where the function is of the form $f(x) = x^{n}$, is given by $f'(x) = nx^{n-1}$. It is a fundamental rule in Calculus and simplifies the process of finding the derivative of polynomial functions, making differentiation more straightforward and efficient.

Example of power rule

Question: Differentiate the function $f(x) = x^{-3}$.

Answer: Simply apply the power rule over here, which states that given a function $f(x)$ we can find derivative $f'(x) = nx^{n-1}$ .

Constant rule

The constant rule in differentiation states that the derivative of any constant value remains zero. When differentiating a function with respect to its variable, constants vanish, as they do not contribute to the rate of change. Thus, the derivative of a constant is always zero. Let's represent a function $f(x) = c$ where c is constant.

Example of constant rule

Question: Differentiate the function $f(x) = 101$

Answer: As there are no terms with $x$ then we can simply say that the derivative of a constant function is 0 i.e. $f'(x) = 0$

Sum and difference rule

The sum rule or difference rule states that the derivative of the sum or differences of two functions is equal to the sum of their individual derivatives:

You can see here for more details of the sum and difference rule.

Product rule

The product rule is a fundamental rule in Calculus used to differentiate the product of two functions. If we have a function $h(x)$ that can be expressed as two functions, $f(x)$ and $g(x)$, the product rule can be expressed as:

You can see here for more details of the product rule.

Quotient rule

The quotient rule states that if you have a function that is a quotient of two functions, such as $h(x)=\frac{f(x)}{g(x)}$ then we can the derivative of such expression by the quotient rule:

You can see here for more details of the quotient rule.

Chain rule

The chain rule states that the derivative of a composite function $y=f(g(x))$ is equal to the derivative of the outer function $f$ with respect to its inner variable $u=g(x)$, multiplied by the derivative of the inner function $g$ with respect to $x$.

You can see here for more details of the chain rule.

Conclusion

Differentiation is a fundamental concept in Calculus that allows us to find the rate of change of a function with respect to its input variable. It plays a crucial role in various fields, such as physics, engineering, economics, and biology.