Product Rule

Differentiation is a fundamental concept in calculus, which is important for understanding how functions change. The derivative of a function gives us the rate of change of that function. One of the important techniques in differentiation is the product rule. This rule is especially useful when you want to differentiate the product of two functions.

Introduction to differentiation

Before we dive deeper into the multiplication rule, let's briefly understand what differentiation means. Consider a function y = f(x). The derivative of this function, represented as f'(x) or dy/dx, represents the rate at which y changes with respect to x.

Differentiation helps find slopes of curves, rates of change in physics, and has a variety of applications in fields such as economics, engineering, and biology.

Understanding the multiplication rule

When a function is the product of two different functions, such as u(x) and v(x), differentiating each function separately and multiplying the results does not give the correct answer. This is where the multiplication rule comes into play.

The product rule says that if you have a function y that is the product of two functions, say:

y = u(x) * v(x)

So the derivative of y with respect to x is:

dy/dx = u(x) * dv/dx + v(x) * du/dx

This formula can also be remembered like this:

(First function * derivative of second function) + (Second function * derivative of first function)

Viewing the product rule

₀ , y ₀ )

In the graph above, suppose the black line represents the product of two functions u(x) and v(x). At any point (x ₀ , y ₀ ) on the graph, you can imagine applying the product rule to find the slope of the tangent to this line.

Why the product rule works

To understand why the multiplication rule works, consider two functions u(x) and v(x). Suppose you want to find the change in their product u(x)*v(x) as x changes by a small amount h. When x changes to (x + h), the new product is:

[u(x) + Δu] * [v(x) + Δv]

Expanding this expression using the distributive property gives:

u(x)v(x) + u(x)Δv + v(x)Δu + ΔuΔv

The changes to the product are as follows:

Δ(u*v) = u(x)Δv + v(x)Δu + ΔuΔv

Dividing the entire equation by h (and assuming h is near zero):

Δ(u*v)/h = u(x)(Δv/h) + v(x)(Δu/h) + (Δu/h)(Δv/h)

As h → 0, both Δu/h and Δv/h approach the derivatives of u and v, respectively. Therefore, the limit of change in the product is given by:

dy/dx = u(x) * dv/dx + v(x) * du/dx

Examples of the multiplication rule

Let's look at how the multiplication rule is applied in practice through some examples.

Example 1

Consider the functions u(x) = x^2 and v(x) = e^x. We want to differentiate the functions:

y = x^2 * e^x

According to the multiplication rule:

dy/dx = (d(x^2)/dx * e^x) + (x^2 * d(e^x)/dx)

The derivative of x^2 is 2x, and the derivative of e^x is e^x. Substituting these values, we get:

dy/dx = (2x * e^x) + (x^2 * e^x)

Combining these terms, we get:

dy/dx = x^2*e^x + 2x*e^x

Example 2

Let's take another example of trigonometric and algebraic functions. Differentiate the function:

y = x * sin(x)

Use the multiplication rule here:

dy/dx = (d(x)/dx * sin(x)) + (x * d(sin(x))/dx)

The derivative of x is 1, and the derivative of sin(x) is cos(x). Substituting these values gives:

dy/dx = (1 * sin(x)) + (x * cos(x))

Thus, the derivative is:

dy/dx = sin(x) + x*cos(x)

Example 3

Differentiate the function y:

y = (2x^3) * ln(x)

Applying the product rule:

dy/dx = (d(2x^3)/dx * ln(x)) + ((2x^3) * d(ln(x))/dx)

The derivative of 2x^3 is 6x^2, and the derivative of ln(x) is 1/x. Substituting these values gives:

dy/dx = (6x^2 * ln(x)) + ((2x^3) * (1/x))

Simplifying the words:

dy/dx = 6x^2*ln(x) + 2x^2

Common mistakes to avoid

When using the product rule, beginners often make some common mistakes:

• Forgetting the rule: Simply multiplying two derivatives without using the rule.
• Wrong derivative: Not finding the derivatives of u(x) or v(x) correctly.
• Combining words incorrectly: Failing to simplify or combine words properly.

Practicing the multiplication rule

Practice is essential to become proficient at applying the multiplication rule. Here are some practice problems you can try:

• Find the derivative of y = (x^3 + 3x) * (cos(x) - 1)
• Differentiate y = (5x^2 - 4) * ln(4x)
• Calculate dy/dx for y = (x + 7) * e^(-x)
• Find the derivative of y = (sec(x)) * (tan(x))
• Differentiate y = sqrt(x) * x^x (Hint: use logarithmic differentiation after applying the multiplication rule)

When practicing, make sure you write down each step clearly, and properly show your application of the multiplication rule.

Conclusion

The multiplication rule is a powerful tool in calculus that allows us to efficiently differentiate the product of two functions. By understanding and memorizing the rule, avoiding common mistakes, and practicing consistently, you can confidently apply this rule to a wide variety of functions. Mastering the multiplication rule, along with other differentiation rules, is an important step in exploring deeper calculus topics such as integration, series, and differential equations.

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