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Chapter 3 | Derivatives
CHAPTER 3 REVIEW
KEY TERMS
acceleration amount of change average rate of change
is the rate of change of the velocity, that is, the derivative of velocity the amount of a function f ( x ) over an interval ⎡ ⎣ x , x + h ⎤
⎦ is f ( x + h )− f ( x )
chain rule differentiable at a differentiable function differentiable on S differentiation higher-order derivative f ( x + h )− f ( a ) b − a the chain rule defines the derivative of a composite function as the derivative of the outer function evaluated at the inner function times the derivative of the inner function the derivative of a constant c multiplied by a function f is the same as the constant multiplied by the derivative: d dx ⎛ ⎝ cf ( x ) ⎞ ⎠ = cf ′( x ) the derivative of a constant function is zero: d dx ( c ) =0, where c is a constant the slope of the tangent line to a function at a point, calculated by taking the limit of the difference quotient, is the derivative gives the derivative of a function at each point in the domain of the original function for which the derivative is defined of a function f ( x ) at a is given by f ( a + h )− f ( a ) h or f ( x )− f ( a ) x − a the derivative of the difference of a function f and a function g is the same as the difference of the derivative of f and the derivative of g : d dx ⎛ ⎝ f ( x )− g ( x ) ⎞ ⎠ = f ′( x )− g ′( x ) a function for which f ′( a ) exists is differentiable at a a function for which f ′( x ) exists is a differentiable function a function for which f ′( x ) exists for each x in the open set S is differentiable on S the process of taking a derivative a derivative of a derivative, from the second derivative to the n th derivative, is called a higher- order derivative constant multiple rule constant rule derivative derivative function difference quotient difference rule implicit differentiation dy dx for a function defined by an equation, accomplished by differentiating both sides of the equation (remembering to treat the variable y as a function) and solving for is a technique for computing is a function f ( x ) over an interval ⎡ ⎣ x , x + h ⎤ ⎦ is
dy dx the rate of change of a function at any point along the function a , also called f ′( a ),
instantaneous rate of change
or the derivative of the function at a
logarithmic differentiation marginal cost marginal profit is a technique that allows us to differentiate a function by first taking the natural logarithm of both sides of an equation, applying properties of logarithms to simplify the equation, and differentiating implicitly is the derivative of the cost function, or the approximate cost of producing one more item is the derivative of the profit function, or the approximate profit obtained by producing and selling one more item is the derivative of the revenue function, or the approximate revenue obtained by selling one more item marginal revenue
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