How To Differentiate Sin 2X?

Understanding the Basics

Before we dive into the differentiation of sin(2x), it's crucial to have a good grasp of some fundamental concepts. At the A-Level Maths level, you should be comfortable with trigonometric functions, their properties, and basic differentiation rules.

Trigonometric Functions: Sin(x) is a trigonometric function that represents the ratio of the length of the side opposite to an angle in a right triangle to the length of the hypotenuse. It oscillates between -1 and 1 as x varies.

Differentiation Rules: You should be familiar with the basic rules of differentiation, such as the power rule, product rule, chain rule, and sum/difference rule.

Now, let's tackle the differentiation of sin(2x) step by step.

Step 1: Identify the Function

In this case, our function is sin(2x). To differentiate it, we need to apply the chain rule because we have a composition of functions: sin(u), where u = 2x.

Step 2: Apply the Chain Rule

The chain rule states that if you have a function f(g(x)), its derivative is f'(g(x)) * g'(x). In our case, f(u) is sin(u), and g(x) is 2x.

Now, we need to find the derivatives of both f(u) and g(x).

Derivative of f(u) = sin(u): You should already know that the derivative of sin(u) is cos(u).

Derivative of g(x) = 2x: The derivative of 2x is simply 2.

Step 3: Combine the Derivatives

Now that we have the derivatives of both f(u) and g(x), we can use the chain rule to find the derivative of sin(2x).

Derivative of sin(2x) = cos(2x) * 2

Step 4: Simplify

To simplify further, multiply cos(2x) by 2:

Derivative of sin(2x) = 2cos(2x)

And there you have it! You've successfully differentiated sin(2x).

Final Thoughts

In A-Level Maths, mastering differentiation is essential for tackling a wide range of mathematical problems, and trigonometric functions like sin(2x) are no exception. Remember that practice makes perfect, so make sure to practice differentiating various functions to become more confident in your skills.

In this blog post, we've broken down the differentiation of sin(2x) into simple steps, emphasizing the application of the chain rule. By following these steps and practicing regularly, you'll be well-prepared to handle more complex differentiation challenges that may come your way in A-Level Maths and beyond.

In conclusion, don't let the intricacies of trigonometric functions intimidate you. With dedication and a solid understanding of differentiation rules, you can differentiate sin(2x) and many other functions with ease. So, keep up the good work, and keep exploring the fascinating world of mathematics!