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Questions and Answers
If f(x) = 3x^2, what is f'(x)?
If f(x) = x^2 sin(x), what is f'(x)?
If f(x) = sin(x) / x, what is f'(x)?
If f(x) = sin(x^2), what is f'(x)?
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If f(x) = x^2 + sin(x), what is f'(x)?
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Study Notes
Derivative Rules
Power Rule
- If
f(x) = x^n, thenf'(x) = nx^(n-1) - Example: If
f(x) = x^2, thenf'(x) = 2x
Product Rule
- If
f(x) = u(x)v(x), thenf'(x) = u'(x)v(x) + u(x)v'(x) - Example: If
f(x) = x^2 sin(x), thenf'(x) = 2x sin(x) + x^2 cos(x)
Quotient Rule
- If
f(x) = u(x)/v(x), thenf'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2 - Example: If
f(x) = sin(x) / x, thenf'(x) = (cos(x) \* x - sin(x)) / x^2
Chain Rule
- If
f(x) = g(h(x)), thenf'(x) = g'(h(x)) \* h'(x) - Example: If
f(x) = sin(x^2), thenf'(x) = 2x cos(x^2)
Sum and Difference Rule
- If
f(x) = u(x) ± v(x), thenf'(x) = u'(x) ± v'(x) - Example: If
f(x) = x^2 + sin(x), thenf'(x) = 2x + cos(x)
These rules can be applied to find the derivative of various functions.
Derivative Rules
Power Rule
- The derivative of
f(x) = x^nisf'(x) = nx^(n-1) - Applies to functions of the form
f(x) = xraised to a powern
Product Rule
- The derivative of
f(x) = u(x)v(x)isf'(x) = u'(x)v(x) + u(x)v'(x) - Applies to functions that are the product of two functions
u(x)andv(x)
Quotient Rule
- The derivative of
f(x) = u(x)/v(x)isf'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2 - Applies to functions that are the quotient of two functions
u(x)andv(x)
Chain Rule
- The derivative of
f(x) = g(h(x))isf'(x) = g'(h(x)) * h'(x) - Applies to composite functions, where a function
gis composed withh(x)
Sum and Difference Rule
- The derivative of
f(x) = u(x) ± v(x)isf'(x) = u'(x) ± v'(x) - Applies to functions that are the sum or difference of two functions
u(x)andv(x)
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Description
Quiz on the rules of derivatives in calculus, including the power rule, product rule, quotient rule, and chain rule.
