Chapter 3 | Derivatives
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d dx ⎞ ⎠ = b g ( x ) g ′( x )ln b • Derivative of the general logarithmic function d dx ⎛ ⎝ log b g ( x ) ⎞ ⎠ = g ′( x ) g ( x )ln b KEY CONCEPTS 3.1 Defining the Derivative • The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. We can calculate it by finding the limit of the difference quotient or the difference quotient with increment h . • The derivative of a function f ( x ) at a value a is found using either of the definitions for the slope of the tangent line. • Velocity is the rate of change of position. As such, the velocity v ( t ) at time t is the derivative of the position s ( t ) at time t . Average velocity is given by v ave = s ( t )− s ( a ) t − a . ⎛ ⎝ b g ( x )
Instantaneous velocity is given by
s ( t )− s ( a ) t − a .
v ( a ) = s ′( a ) = lim t → a
• We may estimate a derivative by using a table of values.
3.2 The Derivative as a Function • The derivative of a function f ( x ) is the function whose value at x is f ′( x ).
• The graph of a derivative of a function f ( x ) is related to the graph of f ( x ). Where f ( x ) has a tangent line with positive slope, f ′( x ) >0. Where f ( x ) has a tangent line with negative slope, f ′( x ) <0. Where f ( x ) has a horizontal tangent line, f ′( x ) =0. • If a function is differentiable at a point, then it is continuous at that point. A function is not differentiable at a point if it is not continuous at the point, if it has a vertical tangent line at the point, or if the graph has a sharp corner or cusp. • Higher-order derivatives are derivatives of derivatives, from the second derivative to the n th derivative.
3.3 Differentiation Rules
• The derivative of a constant function is zero. • 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. • The derivative of a constant c multiplied by a function f is the same as the constant multiplied by the derivative. • 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. • 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. • 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. • The derivative of the quotient of two functions is the derivative of the first function times the second function minus
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