Solving Product Rule Problems with Algebraic Functions Tutorial

📚 Understanding the Product Rule

The product rule is a fundamental concept in calculus that allows us to find the derivative of a function that is the product of two or more functions. In simpler terms, if you have a function like $f(x) = u(x) \cdot v(x)$, the product rule helps you find $f'(x)$.

📜 History and Background

The product rule, along with other core concepts of calculus, was developed independently by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. Leibniz is often credited with the notation we commonly use today.

🔑 Key Principles of the Product Rule

The product rule states that if $f(x) = u(x) \cdot v(x)$, then the derivative $f'(x)$ is given by:

$f'(x) = u'(x)v(x) + u(x)v'(x)$

In words, the derivative of the product is the derivative of the first function times the second function, plus the first function times the derivative of the second function.

✏️ Applying the Product Rule: Algebraic Functions

Let's consider some examples to illustrate how to apply the product rule with algebraic functions.

Example 1:

Find the derivative of $f(x) = (x^2 + 1)(x^3 + 3x)$.

Here, $u(x) = x^2 + 1$ and $v(x) = x^3 + 3x$.

First, find the derivatives of $u(x)$ and $v(x)$:

$u'(x) = 2x$

$v'(x) = 3x^2 + 3$

Now, apply the product rule:

$f'(x) = u'(x)v(x) + u(x)v'(x)$

$f'(x) = (2x)(x^3 + 3x) + (x^2 + 1)(3x^2 + 3)$

Simplify:

$f'(x) = 2x^4 + 6x^2 + 3x^4 + 3x^2 + 3x^2 + 3$

$f'(x) = 5x^4 + 12x^2 + 3$

Example 2:

Find the derivative of $g(x) = (3x - 2)(x^2 + 5)$.

Here, $u(x) = 3x - 2$ and $v(x) = x^2 + 5$.

First, find the derivatives of $u(x)$ and $v(x)$:

$u'(x) = 3$

$v'(x) = 2x$

Now, apply the product rule:

$g'(x) = u'(x)v(x) + u(x)v'(x)$

$g'(x) = (3)(x^2 + 5) + (3x - 2)(2x)$

Simplify:

$g'(x) = 3x^2 + 15 + 6x^2 - 4x$

$g'(x) = 9x^2 - 4x + 15$

💡 Tips for Success

• 🔍 Identify $u(x)$ and $v(x)$: Correctly identifying the two functions is the first crucial step.
• ➕ Apply the Formula: Remember the product rule formula: $f'(x) = u'(x)v(x) + u(x)v'(x)$.
• 📝 Simplify: Always simplify the resulting expression to its simplest form.
• ✅ Double-Check: Ensure you've correctly calculated the derivatives of $u(x)$ and $v(x)$.

✍️ Practice Quiz

Find the derivatives of the following functions:

1. $f(x) = (x^3 + 2x)(x^4 - x)$
2. $g(x) = (2x^2 - 1)(x + 3)$
3. $h(x) = (4x + 5)(x^2 - 2x)$

Solutions:

1. $f'(x) = (3x^2 + 2)(x^4 - x) + (x^3 + 2x)(4x^3 - 1) = 7x^6 + 7x^4 - 4x^3 - 3x^2 - 2$
2. $g'(x) = (4x)(x + 3) + (2x^2 - 1)(1) = 6x^2 + 12x - 1$
3. $h'(x) = (4)(x^2 - 2x) + (4x + 5)(2x - 2) = 12x^2 + 2x - 10$

🌍 Real-World Applications

The product rule isn't just a theoretical concept; it has practical applications in various fields:

• 📈 Economics: Analyzing revenue functions where price and quantity are both functions of another variable.
• 🚀 Physics: Calculating rates of change in systems involving multiple interacting components.
• 🧪 Engineering: Modeling dynamic systems where multiple factors influence the outcome.

🏁 Conclusion

The product rule is a powerful tool for differentiating functions that are products of other functions. By understanding its principles and practicing with examples, you can master this essential calculus technique. Keep practicing, and you'll become more comfortable with its applications!