First Order Derivatives

Calculating first-order derivatives is an important concept in calculus that allows us to determine the slope of a function at a particular point.

This can be useful in a variety of applications, such as optimizing a function or understanding how a function changes as its input changes.

In this article, we will walk through the process of calculating first-order derivatives and provide some examples to help you understand the concept.

First derivatives are obtained by taking the derivative of a given quantity with respect to itself.

$\frac{d}{dx}$ [f(x)]

For example

Evaluate the first derivative of 2x² + sin(x)

Solution

Step 1: Take the given function and apply the notation of differentiation to it and use laws.

$\frac{d}{dx}$ [f(x)] = $\frac{d}{dx}$ [2x² + sin(x)]

$\frac{d}{dx}$ [2x² + sin(x)] = $\frac{d}{dx}$ [2x²] + $\frac{d}{dx}$ [sin(x)]

Step 2: Now differentiate the above expression.

$\frac{d}{dx}$ [2x² + sin(x)] = [4x^2-1] + [cos(x)]

$\frac{d}{dx}$ [2x² + sin(x)] = 4x + cos(x)

Hence, 4x + cos(x) is the first derivative of the given function.

A derivative calculator can also be used to avoid larger calculations for determining the derivative of complex functions in a fraction of a second.