How to find derivative of (x² + 4) · cot(x) ?

 

In this article, we will delve into the step-by-step process of finding the derivative of the function (x² + 4) · cot(x). To do so, we will utilize differentiation rules and techniques to simplify the expression and determine its derivative. Calculus is a branch of mathematics that explores rates of change, and one of its fundamental concepts is differentiation, which helps us understand how a function changes concerning its input. 

Understanding the Problem

Before we begin finding the derivative, let's closely examine the given function:

  • f(x) = (x² + 4) · cot(x)

To find the derivative of this function, we will utilize differentiation rules and techniques, including the product rule and trigonometric identities. The product rule states that if you have a function of the form u(x) · v(x), where both u(x) and v(x) are differentiable, then the derivative can be found using the formula:

  • f'(x) = u'(x) · v(x) + u(x) · v'(x)

Step 1: Identifying u(x) and v(x)

In our case, u(x) = (x² + 4) and v(x) = cot(x). We will need to find the derivatives of both these functions to apply the product rule.

Step 2: Finding u'(x) and v'(x)

To find u'(x), we'll differentiate u(x) with respect to x:

  • u(x) = x² + 4
  • u'(x) = 2x

Now, let's find v'(x) by differentiating v(x) with respect to x:

  • v(x) = cot(x)

To find v'(x), we will use the derivative of cot(x) with respect to x. The derivative of cot(x) is -csc²(x):

  • v'(x) = -csc²(x)

Step 3: Applying the Product Rule

Now that we have u'(x), v'(x), u(x), and v(x), we can apply the product rule to find f'(x):

  • f'(x) = u'(x) · v(x) + u(x) · v'(x)
  • f'(x) = (2x) · (cot(x)) + (x² + 4) · (-csc²(x))

Step 4: Simplifying the Expression

Now, let's simplify the expression further. We can factor out common factors and use trigonometric identities to simplify the expression:

  • f'(x) = 2x · cot(x) - (x² + 4) · csc²(x)

Now, we have successfully found the derivative of the function (x² + 4) · cot(x):

  • f'(x) = 2x · cot(x) - (x² + 4) · csc²(x)

Conclusion

In this comprehensive guide, we have walked through the process of finding the derivative of the function (x² + 4) · cot(x) step by step. By applying the product rule and simplification techniques, we have obtained the final result:

  • f'(x) = 2x · cot(x) - (x² + 4) · csc²(x)

Calculus and differentiation are fundamental concepts in mathematics and play a vital role in various fields such as physics, engineering, economics, and more. Understanding how to find derivatives is essential for analyzing functions and their behavior, and it forms the basis for solving a wide range of real-world problems.

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