The rules of differentiation are a set of fundamental principles used to compute derivatives of mathematical functions. Derivatives play a critical role in calculus, and the rules of differentiation provide a systematic approach to computing them. There are several rules of differentiation, each with its specific application, such as the sum rule, difference rule, power rule, product rule, quotient rule, and chain rule. Understanding these rules is essential for anyone interested in mathematical analysis or problem-solving involving derivatives. In this article, we will explore each of these rules in detail, along with their applications in various real-world scenarios.
Sum Rule
Let $\rm f(x)$ and $\rm g(x)$ be two functions, then according to the sum rule of derivative, the differentiation of their sum is equal to the sum of the differentiation of the individual functions.
\begin{equation} \rm \frac{d}{dx} \left [ f(x) + g(x) \right ] = \frac{d}{dx} f(x) + \frac{d}{dx} g(x) \end{equation}
Difference Rule
Let $\rm f(x)$ and $\rm g(x)$ be two functions, then according to the sum rule of derivative, the differentiation of their difference is equal to the difference of the differentiation of the individual functions.
\begin{equation} \rm \frac{d}{dx} \left [ f(x) - g(x) \right ] = \frac{d}{dx} f(x) - \frac{d}{dx} g(x) \end{equation}
Power Rule
Let $\rm f(x) = x^n$ be any function, then according to power rule of derivative, the differentiation of the function is given by
\begin{equation} \rm \frac{d}{dx} (x)^n = n \cdot x^{n-1} \end{equation}
Product Rule
Let $\rm f(x)$ and $\rm g(x)$ be two functions. Let $\rm f'(x)$ and $\rm g'(x)$ be their first derivatives. Then, according to product rule, the differentiation of their product is given by
\begin{equation} \rm \frac{d}{dx} \left [ \ f(x) \cdot g(x) \ \right ] = g(x) f'(x) + f(x) g'(x) \end{equation}
Quotient Rule
Let $\rm f(x)$ and $\rm g(x)$ be two functions. Let $\rm f'(x)$ and $\rm g'(x)$ be their first derivatives. Then, according to quotient rule, the differentiation of their ratio is given by
\begin{equation} \rm \frac{d}{dx} \left [ \frac{f(x)}{g(x)} \ \right ] = \frac{g(x) f'(x) \ - \ f(x) g'(x)}{ (g(x))^2} \end{equation}
Chain Rule
Let $\rm y = f(u)$ and $\rm u = g(x)$ be two differentiable functions. Then, according to chain rule, the differentiation $\rm \frac{dy}{dx}$ exists and is given by
\begin{equation} \rm \frac{d}{dx} (y) = \frac{dy}{du} \cdot \frac{du}{dx} = f'(u) \frac{du}{dx} \end{equation}