Math Problem Statement
find f''(x) f(x)=ln[(x2+4)(x3+8x−3)2]
Solution
We are given the function . To find the second derivative , we will proceed step-by-step, using the chain rule and product rule as necessary.
Step 1: Simplify the logarithmic expression
We can apply the logarithm property:
and
Thus, the given function becomes:
Step 2: Find the first derivative
Now, differentiate with respect to using the chain rule.
[ f'(x) = \frac{d}{dx}\left[\ln(x^2 + 4)\right] + 2\frac{d}{dx}\left[\ln(x^3 + 8x - 3)\right] ]
Using the chain rule for each term:
The derivatives of the inner functions are:
and
Thus, we have:
Step 3: Find the second derivative
Now we differentiate to find . This will require the quotient rule for both terms. Recall that the quotient rule is:
Let's apply the quotient rule to each term.
First term:
Here, and . Differentiating:
and
Applying the quotient rule:
= \frac{2(x^2 + 4) - 4x^2}{(x^2 + 4)^2} = \frac{-2x^2 + 8}{(x^2 + 4)^2} = \frac{2(4 - x^2)}{(x^2 + 4)^2}$$ #### Second term: $$\frac{d}{dx} \left( 2 \cdot \frac{3x^2 + 8}{x^3 + 8x - 3} \right)$$ Here, $$u = 3x^2 + 8$$ and $$v = x^3 + 8x - 3$$. Differentiating: $$u' = 6x$$ and $$v' = 3x^2 + 8$$ Now applying the quotient rule: $$\frac{d}{dx} \left( \frac{3x^2 + 8}{x^3 + 8x - 3} \right) = \frac{(x^3 + 8x - 3)(6x) - (3x^2 + 8)(3x^2 + 8)}{(x^3 + 8x - 3)^2}$$ This can be simplified, but the result is quite complex, so we will leave it in this form for now. Thus, the second derivative is: $$f''(x) = \frac{2(4 - x^2)}{(x^2 + 4)^2} + 2 \cdot \frac{(x^3 + 8x - 3)(6x) - (3x^2 + 8)(3x^2 + 8)}{(x^3 + 8x - 3)^2}$$ Let me know if you'd like to simplify this further or go over specific steps in more detail. --- Would you like further clarification on this result? Here are five related questions to explore: 1. How do you apply the product and quotient rules in calculus? 2. What are some applications of logarithmic differentiation in solving complex functions? 3. How do higher-order derivatives relate to the curvature of a graph? 4. What techniques simplify complex logarithmic derivatives? 5. How does the chain rule apply to composite functions? **Tip:** When dealing with logarithms, breaking them into simpler components using logarithmic properties often makes differentiation easier.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Differentiation
Logarithmic Functions
Chain Rule
Product Rule
Quotient Rule
Formulas
ln(ab) = ln(a) + ln(b)
ln(b^2) = 2ln(b)
Quotient Rule: d/dx(u/v) = (v * u' - u * v') / v^2
Chain Rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
Theorems
Logarithmic Differentiation Theorem
Quotient Rule Theorem
Chain Rule Theorem
Suitable Grade Level
Undergraduate Calculus
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