Evaluating dsin2x/dx Using the Chain Rule

When evaluating complex derivatives such as dsin2x/dx, the chain rule becomes an essential tool in our mathematical arsenal. The chain rule allows us to break down the differentiation process into simpler steps, making it easier to evaluate even the most complicated functions. In this article, we will explore the underlying process and provide clear, detailed steps to solve such problems effectively.

Understanding the Chain Rule

The chain rule is a fundamental concept in calculus that enables us to differentiate composite functions. It states that if a function ( y ) is a function of ( u ), and ( u ) is a function of ( x ), then the derivative of ( y ) with respect to ( x ) can be given as:

dy/dx (dy/du) * (du/dx)

Applying the Chain Rule to dsin2x/dx

Let's consider the function sin2x. We need to find the derivative of this function with respect to ( x ). Here's a step-by-step guide to evaluating this derivative using the chain rule.

Step 1: Differentiate sin2x with respect to sinx

The first step is to express the derivative of ( sin2x ) in terms of ( sinx ). We start by using the fact that the function ( sin2x ) is a composite function of ( 2x ), and we know that sin2x 2sinx cosx.

Step 2: Apply the chain rule

We will now apply the chain rule, breaking the process into two parts:

Part 1: Differentiating 2sin2x with respect to sinx
We recognize that ( sin2x 2sinx cosx ). Hence, the derivative of ( sin2x ) with respect to ( sinx ) is the derivative of ( 2sinx cosx ) with respect to ( sinx ).

Part 2: Differentiating sinx with respect to x
The derivative of ( sinx ) with respect to ( x ) is a well-known result: d(sinx)/dx cosx.

Step 3: Combine the results

To find the overall derivative, we multiply the results of the two parts derived in steps 1 and 2:

dsin2x/dx (d(sin2x)/d(sinx)) * (d(sinx)/dx)

Substituting the results, we get:

dsin2x/dx (2cosx) * (cosx) 2cos^2x

Conclusion

By breaking down the problem using the chain rule, we have successfully found the derivative of sin2x with respect to ( x ). The process can be summarized as follows:

Recognize ( sin2x 2sinx cosx ). Apply the chain rule to differentiate with respect to ( sinx ) and then ( x ). Multiply the results to get the final derivative.

Additional Tips

When dealing with complex derivatives, always break down the function into simpler parts and apply the chain rule step-by-step. This method ensures accuracy and makes the solution easier to understand.

For more resources on calculus, differentiation, and applications of the chain rule, refer to the following links:

Khan Academy Derivative Rules MathIsFun Chain Rule

Understanding and mastering the chain rule is crucial for advanced calculus and differential equations. With practice and a clear understanding, you can tackle even the most challenging derivative problems with confidence.