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Calculus Product Rule Quiz

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Questions and Answers

What is the product rule in calculus?

The derivative of a product of two functions is the first function multiplied by the derivative of the second function plus the second function multiplied by the derivative of the first function. (correct)

The sum of two functions is equal to their derivatives.

The integral of a product of two functions is the product of their integrals.

The derivative of a product of two functions is the product of their derivatives.

The product rule can only be applied to two differentiable functions.

True

Provide an example of the product rule using functions f(x) = x^2 and g(x) = sin(x).

The derivative is f'(x)g(x) + g'(x)f(x) = 2xsin(x) + x^2cos(x).

If f(x) = x^3 and g(x) = e^x, then using the product rule, the derivative f'(x) * g(x) + g'(x) * f(x) gives _____ as a result.

<p>3x^2 * e^x + x^3 * e^x.</p> Signup and view all the answers

Match the following functions with their derivatives as per product rule applications:

<p>h(x) = x^2 * ln(x) = h'(x) = 2x * ln(x) + x * (1/x) * x^2 k(x) = x * cos(x) = k'(x) = cos(x) + x * -sin(x) m(x) = e^x * x = m'(x) = e^x + x * e^x n(x) = sin(x) * x^3 = n'(x) = 3x^2 * sin(x) + x^3 * cos(x)</p> Signup and view all the answers

Study Notes

Product Rule in Calculus

The product rule is used to find the derivative of the product of two differentiable functions:
- f(x) * g(x)
The formula for the product rule is:
- (f(x) * g(x))' = f'(x) * g(x) + g'(x) * f(x)
The product rule can only be applied to two differentiable functions.

Example of Product Rule

f(x) = x^2, g(x) = sin(x)
- f'(x) = 2x, g'(x) = cos(x)
- Applying the product rule:
  - (x^2 * sin(x))' = 2x * sin(x) + cos(x) * x^2

Derivative using Product Rule

f(x) = x^3, g(x) = e^x
- f'(x) = 3x^2, g'(x) = e^x
- Applying the product rule:
  - f'(x) * g(x) + g'(x) * f(x) = 3x^2 * e^x + e^x * x^3

Matching Functions with Derivatives

This section requires additional information:
- Please provide the functions and their derivatives for matching.

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Description

Test your understanding of the product rule in calculus with this quiz. You will apply the product rule to differentiate functions and match them with their derivatives. Perfect for students looking to strengthen their calculus skills!