Math Problem Statement
If ∫ f ( x ) d x
( x + 3 ) 4 e x + ln ( | x | ) + C , what is f ( x ) ?
Solution
To determine , we differentiate the given integral expression:
Differentiating both sides with respect to :
Step-by-Step Differentiation
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Differentiate : Using the product rule, , where and : [ \frac{d}{dx} \left[ (x+3)^4 e^x \right] = \frac{d}{dx}[(x+3)^4] \cdot e^x + (x+3)^4 \cdot \frac{d}{dx}[e^x] ]
Substituting:
Combine terms: [ \frac{d}{dx} \left[ (x+3)^4 e^x \right] = e^x \left[ 4(x+3)^3 + (x+3)^4 \right] ]
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Differentiate : Recall :
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Differentiate the constant : The derivative of a constant is zero:
Combine Results
Adding all these results gives:
Simplify the expression:
Final Answer:
Would you like a detailed explanation of any specific step?
Related Questions:
- What is the significance of logarithmic differentiation in solving such problems?
- How does the product rule apply to expressions involving both polynomials and exponentials?
- Can this function have singularities? If so, where?
- How can this method be extended to solve for a general ?
- What happens if is replaced with another logarithmic function?
Tip:
When differentiating products or compositions, ensure each rule (e.g., product, chain) is applied consistently to avoid missing terms.
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Math Problem Analysis
Mathematical Concepts
Differentiation
Logarithmic Functions
Exponential Functions
Product Rule
Formulas
Product Rule: d(uv)/dx = u'v + uv'
Derivative of ln(|x|): d/dx[ln(|x|)] = 1/x
Derivative of e^x: d/dx[e^x] = e^x
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Undergraduate Calculus or Advanced High School Calculus
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