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When faced with differentiating a product of two functions, the Product Rule is your best friend. This rule is a key tool in calculus that allows us to find the derivative of two multiplying functions. Imagine this as a way to stitch together the rates of change of both functions to find the overall rate of change for their product. Let's break it down:The Product Rule states that for two functions, say \(f(x)\) and \(g(x)\), the derivative of their product \(f(x)g(x)\) is given by:

• \( (f(x)g(x))' = f'(x)g(x) + f(x)g'(x) \)

To apply this rule, you need to know the derivative of each function separately:

• Step 1: Differentiate \(f(x)\) to get \(f'(x)\)
• Step 2: Differentiate \(g(x)\) to get \(g'(x)\)
• Step 3: Use the Product Rule formula to combine them

This technique is essential when dealing with expressions like our given example \(y=x^{3}(1-x^{2})\). It's straightforward once you understand the steps.

Derivatives are the cornerstone concept of calculus, representing the rate of change or the slope of a function at any point. When it comes to differentiating basic polynomials, such as \(x^3\) or \(1-x^2\), the process is almost mechanical and very straightforward.### Differentiating Each FunctionGiven a function \(y = x^3(1-x^2)\), we first identify and differentiate each of the component functions:

• For \(f(x) = x^3\), using the power rule, the derivative \(f'(x)\) is \(3x^2\), which comes from multiplying the power by the coefficient and reducing the power by one.
• For \(g(x) = 1 - x^2\), the derivative \(g'(x)\) is \(-2x\). The constant term disappears, and the power rule gives us \(-2x\) as the result.

Understanding these derivatives individually is crucial, as we will use them directly when applying the Product Rule.

After applying calculus techniques like differentiation, you're often left with a somewhat messy expression. Simplification involves cleaning up this expression for better readability and utility.### Combining Like TermsSay you've just applied the Product Rule and ended up with the following expression: \[\frac{dy}{dx} = 3x^2 - 3x^4 - 2x^4\]Here, the term \(-3x^4\) and \(-2x^4\) are like terms — they both contain \(x^4\).### Process of Simplification

• **Identify Like Terms**: In this case, \(-3x^4\) and \(-2x^4\) add up to \(-5x^4\).
• **Combine the Terms**: This yields a more straightforward equation like \(3x^2 - 5x^4\).

Simplification makes expressions easier to interpret or further manipulate, an essential skill in solving and analyzing calculus problems effectively.