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The product rule in calculus is a crucial theorem for finding the derivative of two multiplied functions. It states that if y = uv, the derivative dy/dx is u(dv/dx) + v(du/dx). This rule is vital for differentiating products of functions, including trigonometric, polynomial, and logarithmic functions. Examples provided illustrate its application in various mathematical scenarios, enhancing understanding and proficiency.

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## Definition of the Product Rule

### Fundamental theorem in calculus

The product rule is a fundamental theorem in calculus for finding the derivative of the product of two functions

### Importance of memorizing the rule

It is imperative for students to memorize this rule as it is often not included in examination formula sheets

### Application in mathematics and science

The product rule is essential for differentiating products of functions and is widely used in various fields of mathematics and science

## Formula of the Product Rule

### Derivative of a product of two functions

If \( y = uv \), where \( u \) and \( v \) are differentiable functions of \( x \), then the derivative of \( y \) with respect to \( x \) is \( \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \)

### Function notation

In function notation, the product rule is expressed as \( f'(x) = g(x)h'(x) + h(x)g'(x) \), where \( f(x) = g(x)h(x) \)

### Use in abstract functions and implicit definitions

The product rule allows for a clear and concise representation of the rule, facilitating its application in calculus for abstract functions or when functions are defined implicitly

## Examples of the Product Rule

### Product of algebraic and exponential functions

The product rule can be applied to functions such as \( y = x^2e^x \) to find the derivative \( \frac{dy}{dx} = x^2e^x + 2xe^x \)

### Product of trigonometric and algebraic functions

The product rule can be used to differentiate functions like \( y = x\sin(x) \) to find \( \frac{dy}{dx} = x\cos(x) + \sin(x) \)

### Product of polynomial and logarithmic functions

The product rule can also be applied to functions such as \( f(x) = x^3(x + 1) \) and \( y = x\ln(x) \) to find their derivatives

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