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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The product rule is a fundamental theorem in calculus for finding the derivative of the product of two functions
It is imperative for students to memorize this rule as it is often not included in examination formula sheets
The product rule is essential for differentiating products of functions and is widely used in various fields of mathematics and science
If \( y = uv \), where \( u \) and \( v \) are differentiable functions of \( x \), then the derivative of \( y \) with respect to \( x \) is \( \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \)
In function notation, the product rule is expressed as \( f'(x) = g(x)h'(x) + h(x)g'(x) \), where \( f(x) = g(x)h(x) \)
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
The product rule can be applied to functions such as \( y = x^2e^x \) to find the derivative \( \frac{dy}{dx} = x^2e^x + 2xe^x \)
The product rule can be used to differentiate functions like \( y = x\sin(x) \) to find \( \frac{dy}{dx} = x\cos(x) + \sin(x) \)
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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Product Rule Application
Apply to functions as products; differentiate g(x)h(x) using g(x)h'(x) + h(x)g'((x)).
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Implicit Differentiation Use
Use product rule for functions defined implicitly, not explicitly, to find derivatives.
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Define the product rule for differentiation.
Product rule: derivative of u*v = u'v + uv', where u and v are functions of x, and ' denotes differentiation.
