Anish Singhal
Last Activity: 7 Years ago
To find the derivative of the function , we can apply the product rule and the chain rule of differentiation. This function consists of multiple components, so let’s break it down step by step.
Understanding the Components
The function can be viewed as the product of two parts: and . Therefore, we can express as:
Applying the Product Rule
According to the product rule, the derivative of a product is given by:
We need to compute and separately.
Finding
To differentiate , we again use the product rule because it’s a product of and :
Let:
Then:
Here:
- (the derivative of )
- (the derivative of )
Plugging these into the product rule yields:
Finding
Next, we differentiate . This requires the chain rule:
Let , so . The derivative is:
Combining the Derivatives
Now we can substitute and back into the product rule:
Substituting our values:
This simplifies to:
Final Expression for the Derivative
The final expression for the derivative is:
This gives us a clear understanding of how the function changes with respect to . Each component contributes to the overall rate of change, and by breaking it down using calculus rules, we can find the derivative effectively.