Math Problem Statement
Solution
We are tasked with finding the first and second derivatives of the function:
Step 1: Simplify the function
First, simplify the given function by dividing each term in the numerator by :
This simplifies to:
Step 2: First derivative
To find the first derivative, , apply the power rule to each term:
[ y' = \frac{d}{dx} [4x^2] + \frac{d}{dx} [7x^{-1}] ]
Now, differentiate each term:
Simplifying:
Step 3: Second derivative
To find the second derivative, , differentiate :
[ y'' = \frac{d}{dx} [8x] - \frac{d}{dx} \left[\frac{7}{x^2}\right] ]
For , the derivative is 8.
For , use the power rule again:
Thus, the second derivative is:
Final results:
- First derivative:
- Second derivative:
Would you like more details on any step, or have any further questions?
Here are 5 related questions to further expand:
- How would the derivatives change if the power of in the original function were higher or lower?
- Can you apply the product or quotient rule directly to this problem without simplifying first?
- How does the power rule for differentiation work when applied to negative exponents?
- What happens to the behavior of the function as approaches 0 or infinity?
- How would you interpret the second derivative physically or geometrically?
Tip: When simplifying functions before differentiating, breaking down terms can often help make the application of basic rules easier and less prone to error.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Power Rule
Simplifying Expressions
Formulas
Power rule: d/dx [x^n] = n * x^(n-1)
First and second derivatives
Theorems
Differentiation using Power Rule
Suitable Grade Level
High School - Grades 11-12
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