Math Problem Statement
derivative
Solution
The problem asks for the derivative of the function , and then to evaluate .
Step 1: Simplify the Function
Before differentiating, it will be easier to express the function in terms of powers of :
- Recall that , so:
- The second term is . Using , we have: So the function becomes:
Step 2: Differentiate
Now we differentiate term by term using the power rule:
- The derivative of is:
- The derivative of is:
So, the derivative of is:
Step 3: Find
Now we substitute into the derivative expression:
First, calculate the powers of 4:
Now, substitute these values:
Next, find a common denominator to simplify:
Thus, .
Final Answer:
.
Would you like more details or have any questions?
Here are five related questions:
- How do you differentiate functions with fractional exponents?
- What are some common applications of the power rule in real-world problems?
- How does the chain rule apply to expressions involving square roots?
- What is the difference between differentiating polynomials and rational functions?
- How can you simplify complex functions before applying derivative rules?
Tip: Always simplify expressions before differentiating, as it makes the process easier and reduces the chance of errors.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Power Rule
Formulas
Power Rule: d/dx[x^n] = n*x^(n-1)
Derivative of a constant multiple
Theorems
Power Rule
Suitable Grade Level
Grades 11-12