Chapter 3 | Derivatives
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population growth rate power rule is the derivative of the population with respect to time the derivative of a power function is a function in which the power on x becomes the coefficient of the term and the power on x in the derivative decreases by 1: If n is an integer, then d dx x n = nx n −1 the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function: d dx ⎛ ⎝ f ( x ) g ( x ) ⎞ ⎠ = f ′( x ) g ( x )+ g ′( x ) f ( x ) product rule the derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function: d dx ⎛ ⎝ f ( x ) g ( x ) ⎞ ⎠ = f ′( x ) g ( x )− g ′( x ) f ( x ) ⎛ ⎝ g ( x ) ⎞ ⎠ 2 quotient rule speed sum rule is the absolute value of velocity, that is, | v ( t ) | is the speed of an object at time t whose velocity is given by v ( t ) the derivative of the sum of a function f and a function g is the same as the sum of the derivative of f and the derivative of g : d dx ⎛ ⎝ f ( x )+ g ( x ) ⎞ ⎠ = f ′( x )+ g ′( x ) KEY EQUATIONS • Difference quotient Q = f ( x )− f ( a ) x − a • Difference quotient with increment h Q = f ( a + h )− f ( a ) a + h − a = f ( a + h )− f ( a ) h • Slope of tangent line m tan = lim x → a f ( x )− f ( a ) x − a m tan = lim h →0 f ( a + h )− f ( a ) h • Derivative of f ( x ) at a f ′( a ) = lim x → a f ( x )− f ( a ) x − a f ′( a ) = lim h →0 f ( a + h )− f ( a ) h • Average velocity v a ve = s ( t )− s ( a ) t − a • Instantaneous velocity v ( a ) = s ′( a ) = lim t → a s ( t )− s ( a ) t − a • The derivative function f ′( x ) = lim h →0 f ( x + h )− f ( x ) h • Derivative of sine function d dx (sin x ) =cos x • Derivative of cosine function d dx (cos x ) =−sin x • Derivative of tangent function
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