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Questions and Answers
What is the derivative of $x^n?$
What is the derivative of a constant?
What is the derivative of $k * f(x)$?
What is the derivative of f(x) + g(x)?
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What is the notation for the derivative of y with respect to x?
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Study Notes
Rules for Differentiation
- Determining the derivative of a function from first principles requires a long calculation and is prone to mistakes.
- The general rule for differentiation is:
d/dx [x^n] = nx^(n-1), wherenis a real number andnis not equal to 0.
Derivative Rules
- Derivative of a constant:
d/dx [k] = 0. - Derivative of a constant multiplied by a function:
d/dx [k * f(x)] = k * d/dx [f(x)]. - Derivative of a sum:
d/dx [f(x) + g(x)] = d/dx [f(x)] + d/dx [g(x)]. - Derivative of a difference:
d/dx [f(x) - g(x)] = d/dx [f(x)] - d/dx [g(x)].
When to Use the Rules for Differentiation
- Use the rules for differentiation when the question does not specify how to determine the derivative.
- Use first principles if specifically requested or if asked to differentiate using the definition of a derivative.
Notation
- Alternative notations for the derivative include:
f'(x),y',dy/dx,df/dx,d/dx [f(x)],Df(x), andD_xy. -
Dandd/dxare called differential operators because they indicate the operation of differentiation. -
dy/dxmeansydifferentiated with respect tox.
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Description
Learn the rules for differentiation, including the general rule for finding the derivative of a function, making it easier to calculate derivatives.
