What is the derivative of a complex function?

In summary, the derivative of a long function is the slope of the tangent line at a specific point on the function, and can be calculated using various rules such as the power rule, product rule, quotient rule, or chain rule. Its purpose is to analyze the behavior of the function and understand its rate of change. The derivative can be negative, indicating a decrease in the function, and it relates to the original function by representing its slope at each point.
  • #1
PrudensOptimus
641
0
What is the derivative of

y = (x^3/2)(sinxcosx)^2

is it

(x^3/2)(2sinxcosx)(cos^2x - sin^2x) + (3sqrt(x)/2)(sinxcosx)^2 ?
 
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  • #2
yes, I get that answer too
but the derivative becomes easier and neater if you notice that sinxcosx=1/2sin2x
 
  • #3


Yes, that is the correct derivative of the given function. The first term in your answer is the product rule applied to the first two factors, and the second term is the derivative of the third factor using the chain rule. Well done!
 

1. What is the definition of a derivative of a long function?

The derivative of a long function is the slope of the tangent line at a specific point on the function. It represents the rate of change of the function at that point.

2. How is the derivative of a long function calculated?

The derivative of a long function can be calculated using the power rule, product rule, quotient rule, or chain rule.

3. What is the purpose of finding the derivative of a long function?

Finding the derivative of a long function allows us to analyze the behavior of the function, such as identifying critical points, finding the maximum and minimum values, and understanding the function's rate of change.

4. Can the derivative of a long function be negative?

Yes, the derivative of a long function can be negative. This indicates that the function is decreasing at that point.

5. How does the derivative of a long function relate to the original function?

The derivative of a long function is a new function that represents the slope of the original function at each point. This means that the derivative can tell us how the original function is changing at any given point.

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