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Questions and Answers
What is the derivative of the function f(x) = (x^3 + sin(x))/(cos(x))?
What is the derivative of the function f(x) = (x^3 + sin(x))/(cos(x))?
Which of the following is the derivative of the function f(x) = x^2 * e^x * sin(x) ?
Which of the following is the derivative of the function f(x) = x^2 * e^x * sin(x) ?
Find the derivative of the function f(x) = (x^2 + 1) / (x^3 + 1).
Find the derivative of the function f(x) = (x^2 + 1) / (x^3 + 1).
If f(x) = 3x^2 + 2x and g(x) = x^3 - 1, what is the derivative of h(x) = f(x) * g(x)?
If f(x) = 3x^2 + 2x and g(x) = x^3 - 1, what is the derivative of h(x) = f(x) * g(x)?
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What is the derivative of the function f(x) = (x^2 + 1) / (x^2 - 1) ?
What is the derivative of the function f(x) = (x^2 + 1) / (x^2 - 1) ?
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Find the derivative of the function f(x) = sin(x) / cos(x)
Find the derivative of the function f(x) = sin(x) / cos(x)
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Find the derivative of f(x) = (x^2 + 1)(x^3 - 1) using the product rule
Find the derivative of f(x) = (x^2 + 1)(x^3 - 1) using the product rule
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Find the derivative of the function f(x) = (x^2 + 1) / (x^2 - 1) using the quotient rule
Find the derivative of the function f(x) = (x^2 + 1) / (x^2 - 1) using the quotient rule
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What is the derivative of the function f(x) = (1 / (x^2 + 1)) using the reciprocal rule?
What is the derivative of the function f(x) = (1 / (x^2 + 1)) using the reciprocal rule?
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Find the derivative of f(x) = 1 / x^3
Find the derivative of f(x) = 1 / x^3
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Study Notes
Derivatives of Sums and Products of Functions
- Derivatives of sums and products of functions can be computed using simple rules, avoiding the need to calculate complicated limits.
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Derivatives of Sums: The derivative of the sum of two functions is equal to the sum of the derivatives of each function.
- This rule stems from the definition of the derivative, involving the limit of a difference quotient.
- The difference quotient of the sum can be rewritten as the sum of difference quotients, leading to the sum of derivatives.
- Example: The derivative of f(x) = x + ex is f'(x) = 1 + ex because the derivative of x is 1 and the derivative of ex is ex.
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Derivatives of Products: The derivative of the product of two functions is equal to the derivative of the first function times the second function plus the first function times the derivative of the second function.
- This rule is demonstrated using the limit definition, by strategically adding and subtracting a specific term to manipulate the difference quotient.
- The manipulation allows the difference quotient to be expressed as the sum of two terms, each representing a derivative.
- Example: The derivative of f(x) = x2 * sin(x) is f'(x) = 2x * sin(x) + x2 * cos(x) since the derivative of x2 is 2x, the derivative of sin(x) is cos(x), and these terms are applied according to the product rule.
- Derivative of a Reciprocal: The derivative of the reciprocal of a function is the negative of the derivative of the function divided by the square of the original function.
- Derivative of a Ratio: The derivative of the ratio between two functions is equal to the derivative of the numerator multiplied by the denominator minus the numerator multiplied by the derivative of the denominator, all divided by the square of the denominator.
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Description
Explore the rules governing the derivatives of sums and products of functions. This quiz covers the fundamental concepts and examples that illustrate how to calculate derivatives efficiently. Test your understanding of these essential calculus principles.
