Rules for finding derivatives

Calculus

Derivatives of some functions are well-known. Using such well-known derivatives with some simple rules explained in this article, it is easy to arrive at the derivative of more complicated functions.

Prerequisites

To understand the rules for finding derivaties, we recommend familiarity with the concepts in

Follow the above links to first get acquainted with the corresponding concepts.

Rules for computing derivatives

Some easy rules apply when dealing with functions of functions. A brief discussion follows here.

  • The Sum rule helps in calculating the derivative of the sum of two functions, if they are differentiable.

$$ \frac{d (f(x) + g(x))}{dx} = \frac{df(x)}{dx} + \frac{dg(x)}{dx} $$

  • The Product rule helps in calculating the derivative of the product of two functions, if they are differentiable.

$$ \frac{d (f(x) g(x))}{dx} = g(x) \frac{df(x)}{dx} + f(x) \frac{dg(x)}{dx} $$

  • The Quotient rule is used to calculate the derivative of the ratio of two differentiable functions.

$$ \frac{d \left(\frac{f(x)}{g(x)} \right)}{dx} = \frac{\frac{df(x)}{dx} g(x) - f(x) \frac{dg(x)}{dx}}{g(x)^2} $$

  • The Chain rule is used to compute the derivative of a function of a function. If we use the symbol \( y = g(x) \), and if \( f'(y) \) and \( g'(x) \) exists, then we can write

$$ \frac{d (f(g(x)) }{dx} = \frac{df(y)}{dy} \frac{dy}{dx}$$

More generally, one can write

These rules are the building blocks for computing derivatives of complicated functions from the derivatives of well-known simpler functions.

Where to next?

Now that you are an expert in derivatives, explore the counterpart to derivatives — integrals.

Already a calculus expert? Check out comprehensive courses on multivariate calculus, machine learning or deep learning

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