Derivative Of F'(x) = (3x²-3x+5)² (2x²-3)

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Hey guys! Let's break down how to find the derivative of the function f'(x) = (3x²-3x+5)² (2x²-3). This problem involves the product rule and the chain rule, so buckle up, and let's dive in!

Understanding the Problem

Before we start, let's clarify what we're trying to do. We have a function, f'(x), which is the product of two expressions: (3x²-3x+5)² and (2x²-3). Our mission, should we choose to accept it, is to find the derivative of this function, often denoted as f''(x). This will tell us how the rate of change of f'(x) behaves.

The product rule is essential here because we have two functions multiplied together. The product rule states that if you have a function h(x) = u(x)v(x), then its derivative is h'(x) = u'(x)v(x) + u(x)v'(x). In our case, u(x) = (3x²-3x+5)² and v(x) = (2x²-3).

Additionally, we'll need the chain rule to differentiate u(x) because it's a composite function (a function within a function). The chain rule states that if you have a function g(x) = f(u(x)), then its derivative is g'(x) = f'(u(x)) * u'(x). This means we'll differentiate the outer function, keeping the inner function as is, and then multiply by the derivative of the inner function. It sounds complex, but we'll take it step by step.

So, to recap, we need to:

  1. Identify u(x) and v(x).
  2. Find u'(x) using the chain rule.
  3. Find v'(x).
  4. Apply the product rule to find f''(x).

Step-by-Step Solution

Step 1: Identify u(x) and v(x)

As mentioned earlier, let's define our functions:

  • u(x) = (3x² - 3x + 5)²
  • v(x) = (2x² - 3)

Step 2: Find u'(x) using the Chain Rule

To find u'(x), we'll use the chain rule. Let's break down u(x) into an outer function and an inner function.

  • Outer function: f(w) = w²
  • Inner function: w(x) = 3x² - 3x + 5

First, differentiate the outer function:

  • f'(w) = 2w

Now, differentiate the inner function:

  • w'(x) = 6x - 3

Apply the chain rule: u'(x) = f'(w(x)) * w'(x)

  • u'(x) = 2(3x² - 3x + 5) * (6x - 3)
  • u'(x) = (12x - 6)(3x² - 3x + 5)

Step 3: Find v'(x)

Differentiating v(x) = 2x² - 3 is straightforward using the power rule:

  • v'(x) = 4x

Step 4: Apply the Product Rule to Find f''(x)

Now we have everything we need to apply the product rule:

  • f''(x) = u'(x)v(x) + u(x)v'(x)

Substitute the values we found:

Now, let's simplify this expression. First, expand the terms:

f''(x) = (12x - 6)(3x² - 3x + 5)(2x² - 3) + 4x(3x² - 3x + 5)²

Expanding (12x - 6)(3x² - 3x + 5) gives:

12x(3x² - 3x + 5) - 6(3x² - 3x + 5) = 36x³ - 36x² + 60x - 18x² + 18x - 30 = 36x³ - 54x² + 78x - 30

So, the first term becomes:

(36x³ - 54x² + 78x - 30)(2x² - 3) = 72x⁵ - 108x⁴ + 156x³ - 60x² - 108x³ + 162x² - 234x + 90 = 72x⁵ - 108x⁴ + 48x³ + 102x² - 234x + 90

Now, let's expand the second term, 4x(3x² - 3x + 5)²:

First, expand (3x² - 3x + 5)²:

(3x² - 3x + 5)(3x² - 3x + 5) = 9x⁴ - 9x³ + 15x² - 9x³ + 9x² - 15x + 15x² - 15x + 25 = 9x⁴ - 18x³ + 39x² - 30x + 25

Then, multiply by 4x:

4x(9x⁴ - 18x³ + 39x² - 30x + 25) = 36x⁵ - 72x⁴ + 156x³ - 120x² + 100x

Finally, combine the two expanded terms:

f''(x) = (72x⁵ - 108x⁴ + 48x³ + 102x² - 234x + 90) + (36x⁵ - 72x⁴ + 156x³ - 120x² + 100x)

Combine like terms:

f''(x) = 108x⁵ - 180x⁴ + 204x³ - 18x² - 134x + 90

Final Answer

Therefore, the first derivative of f'(x) = (3x² - 3x + 5)² (2x² - 3) is:

f''(x) = 108x⁵ - 180x⁴ + 204x³ - 18x² - 134x + 90

Key Concepts Used

  • Product Rule: This rule is used to find the derivative of a function that is the product of two other functions.
  • Chain Rule: This rule is used to find the derivative of a composite function (a function within a function).
  • Power Rule: This rule is used to differentiate terms of the form axⁿ.

Tips for Solving Similar Problems

  1. Break It Down: Always identify the individual functions and how they are combined (product, quotient, composition).
  2. Apply Rules Methodically: Use the product, quotient, and chain rules step-by-step to avoid errors.
  3. Simplify: After applying the rules, simplify the expression to get a cleaner and more manageable result.
  4. Practice: The more you practice, the more comfortable you'll become with these rules.

Common Mistakes to Avoid

  • Forgetting the Chain Rule: When differentiating composite functions, always remember to multiply by the derivative of the inner function.
  • Incorrectly Applying the Product Rule: Ensure you correctly identify u(x), v(x), u'(x), and v'(x) before applying the rule.
  • Algebraic Errors: Be careful when expanding and simplifying expressions. A small mistake can lead to a completely wrong answer.

Real-World Applications

Derivatives aren't just abstract math concepts; they have numerous applications in the real world. Here are a few examples:

  • Physics: Derivatives are used to calculate velocity and acceleration. For example, if you know the position of an object as a function of time, you can find its velocity by taking the derivative of the position function.
  • Engineering: Derivatives are used to optimize designs and processes. For example, engineers might use derivatives to find the maximum stress on a beam or the minimum cost of a manufacturing process.
  • Economics: Derivatives are used to analyze marginal cost and revenue. For example, economists might use derivatives to determine the optimal production level for a company.
  • Computer Graphics: Derivatives are used to create smooth curves and surfaces in computer graphics. For example, game developers might use derivatives to create realistic animations.

Conclusion

Finding the derivative of f'(x) = (3x²-3x+5)² (2x²-3) involves a combination of the product rule and the chain rule. By breaking down the problem into smaller steps and applying these rules methodically, we can find the derivative and simplify it to obtain f''(x) = 108x⁵ - 180x⁴ + 204x³ - 18x² - 134x + 90. Keep practicing, and you'll master these concepts in no time! Happy calculating!