How to find derivative of sin(x) · cos(x) ?

In this article, we will explore how to find the derivative of the product of two trigonometric functions, sin(x) and cos(x), and discover that the answer is indeed cos2(x). Calculus is a fundamental branch of mathematics that deals with the concepts of change and motion. It provides powerful tools for understanding how functions behave, and one of the key operations in calculus is finding derivatives. Derivatives measure the rate at which a function changes with respect to its input, and they have numerous applications in science, engineering, and everyday life. 

Understanding Trigonometric Functions

Before we dive into finding the derivative of sin(x) * cos(x), it's essential to have a solid grasp of the trigonometric functions involved.

Sine Function (sin(x)):

The sine function, denoted as sin(x), represents the ratio of the length of the side opposite an angle in a right triangle to the length of the hypotenuse. In the context of the unit circle, sin(x) is the y-coordinate of a point on the circle when the angle x is measured from the positive x-axis in a counterclockwise direction. The sine function oscillates between -1 and 1 as x varies.

Cosine Function (cos(x)):

The cosine function, denoted as cos(x), represents the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle. Similarly, in the unit circle context, cos(x) is the x-coordinate of a point on the unit circle when the angle x is measured from the positive x-axis in a counterclockwise direction. The cosine function also oscillates between -1 and 1 as x varies.

Finding the Derivative of sin(x) * cos(x)

To find the derivative of sin(x) * cos(x), we can use the product rule, one of the fundamental rules of differentiation in calculus. The product rule states that if you have two functions, u(x) and v(x), then the derivative of their product is given by: d(uv)/dx = u'v + uv'

In our case, u(x) is sin(x), and v(x) is cos(x). We'll start by finding the derivatives of these functions:

Derivative of sin(x):

The derivative of sin(x) with respect to x is cos(x). This result is one of the well-known properties of trigonometric functions.

Derivative of cos(x):

The derivative of cos(x) with respect to x is -sin(x). Again, this is a well-established property of trigonometric functions.

Now that we have the derivatives of sin(x) and cos(x), we can apply the product rule:
  • d(sin(x) * cos(x))/dx = (cos(x)) * cos(x) + (sin(x)) * (-sin(x))
Simplify this expression:
  • cos^2(x) - sin^2(x)
Using the trigonometric identity cos^2(x) - sin^2(x) = cos2(x), we can rewrite cos^2(x) - sin^2(x)
  • cos2(x)
So, the derivative of sin(x) * cos(x) is cos2(x).

Conclusion

In this article, we've explored the process of finding the derivative of sin(x) * cos(x) using the product rule in calculus. The correct derivative is cos2(x), which is a valuable result in calculus and has applications in various fields of science and engineering. Understanding how to find derivatives of trigonometric functions and applying fundamental rules like the product rule is essential for solving more complex problems in calculus and related areas of mathematics.

Popular posts from this blog

How to find derivative of x⁴ - 3x² - 7x + 4√x - 5 ?

Image
  In this article, we will delve into the process of finding the derivative of a polynomial function that includes both traditional polynomial terms and a square root term. Specifically, we will work through the steps to find the derivative of the function f(x) = x⁴ - 3x² - 7x + 4√x - 5. Calculus, a branch of mathematics that explores rates of change and the accumulation of quantities, is an essential tool in various fields, from physics to economics. One fundamental concept in calculus is finding the derivative of a function, which provides information about how the function changes with respect to its independent variable.  Understanding Basic Differentiation Rules Before we tackle the given function, let's review some basic rules of differentiation that will be useful: Power Rule:  The derivative of x^n with respect to x, where n is a constant, is n*x^(n-1). Constant Rule:  The derivative of a constant C (where C is any real number) is 0. Sum/Difference Rule:  The derivative of
Read more

How to find derivative of 3x² + ²⁄ₓ + 7 - 4tax(x) + 5sec(x) ?

Image
  In this article, we will walk through the process of finding the derivative of a function that incorporates polynomial terms, a fractional term, and trigonometric functions. Specifically, we will explore how to find the derivative of the function: f(x) = 3x² + ²⁄ₓ + 7 - 4tan(x) + 5sec(x).  Calculus, a branch of mathematics that explores the concepts of rates of change and accumulation, is a foundational tool in various scientific and engineering fields. Among its essential concepts is finding the derivative of a function, which provides insights into how the function changes concerning its independent variable.  Understanding Basic Differentiation Rules Before we delve into finding the derivative of this complex function, let's review some fundamental rules of differentiation: Power Rule: The derivative of x^n with respect to x, where n is a constant, is n * x^(n-1). Constant Rule: The derivative of a constant C (where C is any real number) is 0. Sum/Difference Rule : The deriv
Read more

How to find derivative of 4x⁵ - 7x² + 2sin(x) - 4cos(x) - 1 ?

Image
In this article, we will explore the process of finding the derivative of a composite function that includes polynomial terms, sine, and cosine functions. Specifically, we will work through the steps to find the derivative of the function f(x) = 4x⁵ - 7x² + 2sin(x) - 4cos(x) - 1.Calculus, the branch of mathematics that deals with rates of change and accumulation, is a crucial tool in various scientific and engineering disciplines. One of the fundamental concepts in calculus is finding the derivative of a function, which describes how the function's values change as the independent variable changes.  Understanding Basic Differentiation Rules Before we dive into finding the derivative of the given function, let's review some fundamental rules of differentiation that will be essential for our calculations: Power Rule:  The derivative of x^n with respect to x, where n is a constant, is n * x^(n-1). Constant Rule:  The derivative of a constant C (where C is any real number) is 0. Su
Read more