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3.1 Derivative of a Function. We write:. There are many ways to write the derivative of. is called the derivative of at . “The derivative of f with respect to x is …”. 3.1 Derivative of a Function. “the derivative of f with respect to x”. “f prime x”. or. “y prime”.
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3.1 Derivative of a Function We write: There are many ways to write the derivative of is called the derivative of at . “The derivative of f with respect to x is …”
3.1 Derivative of a Function “the derivative of f with respect to x” “f prime x” or “y prime” “the derivative of y with respect to x” or “dee why dee ecks” “the derivative of f with respect to x” or “dee eff dee ecks” “the derivative of f of x” “dee dee ecks uv eff uv ecks” or
3.1 Derivative of a Function Note: dx does not mean d times x ! dy does not mean d times y !
3.1 Derivative of a Function does not mean ! does not mean ! Note: (except when it is convenient to think of it as division.) (except when it is convenient to think of it as division.)
3.1 Derivative of a Function does not mean times ! Note: (except when it is convenient to treat it that way.)
3.1 Derivative of a Function The derivative is the slope of the original function. The derivative is defined at the end points of a function on a closed interval.
3.1 Derivative of a Function A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points.
3.2 Differentiability corner cusp discontinuity vertical tangent To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at:
3.2 Differentiability Most of the functions we study in calculus will be differentiable.
3.2 Differentiability If f has a derivative at x = a, then f is continuous at x = a. There are two theorems on page 110: Since a function must be continuous to have a derivative, if it has a derivative then it is continuous.
3.2 Differentiability If a and b are any two points in an interval on which f is differentiable, then takes on every value between and . Between a and b, must take on every value between and . Intermediate Value Theorem for Derivatives
3.3 Rules for Differentiation The derivative of a constant is zero. If the derivative of a function is its slope, then for a constant function, the derivative must be zero. example:
3.3 Rules for Differentiation We saw that if , . power rule This is part of a pattern. examples:
3.3 Rules for Differentiation constant multiple rule: examples:
3.3 Rules for Differentiation constant multiple rule: sum and difference rules: (Each term is treated separately)
3.3 Rules for Differentiation Horizontal tangents occur when slope = zero. Find the horizontal tangents of: Substituting the x values into the original equation, we get: (The function is even, so we only get two horizontal tangents.)
3.3 Rules for Differentiation First derivative (slope) is zero at:
3.3 Rules for Differentiation product rule: Notice that this is not just the product of two derivatives. This is sometimes memorized as:
3.3 Rules for Differentiation product rule: add and subtract u(x+h)v(x) in the denominator Proof
3.3 Rules for Differentiation quotient rule: or
3.3 Rules for Differentiation is the first derivative of y with respect to x. is the second derivative. is the third derivative. is the fourth derivative. Higher Order Derivatives: (y double prime) We will learn later what these higher order derivatives are used for.
3.3 Rules for Differentiation Suppose u and v are functions that are differentiable at x = 3, and that u(3) = 5, u’(3) = -7, v(3) = 1, and v’(3)= 4. Find the following at x = 3 :
3.4 Velocity and other Rates of Change B distance (miles) A time (hours) (The velocity at one moment in time.) Consider a graph of displacement (distance traveled) vs. time. Average velocity can be found by taking: The speedometer in your car does not measure average velocity, but instantaneous velocity.
3.4 Velocity and other Rates of Change Velocity is the first derivative of position. Acceleration is the second derivative of position.
3.4 Velocity and other Rates of Change Gravitational Constants: Speed is the absolute value of velocity. Example: Free Fall Equation
3.4 Velocity and other Rates of Change Acceleration is the derivative of velocity. example: If distance is in: Velocity would be in: Acceleration would be in:
3.4 Velocity and other Rates of Change distance time acc neg vel pos & decreasing acc neg vel neg & decreasing acc zero vel neg & constant acc zero vel pos & constant acc pos vel neg & increasing velocity zero acc pos vel pos & increasing acc zero, velocity zero
3.4 Velocity and other Rates of Change Average rate of change = Instantaneous rate of change = Rates of Change: These definitions are true for any function. ( x does not have to represent time. )
3.4 Velocity and other Rates of Change For tree ring growth, if the change in area is constant then dr must get smaller as r gets larger. For a circle: Instantaneous rate of change of the area with respect to the radius.
3.4 Velocity and other Rates of Change from Economics: Marginal cost is the first derivative of the cost function, and represents an approximation of the cost of producing one more unit.
3.4 Velocity and other Rates of Change The actual cost is: Example 13: Suppose it costs: to produce x stoves. If you are currently producing 10 stoves, the 11th stove will cost approximately: marginal cost actual cost
3.4 Velocity and other Rates of Change Note that this is not a great approximation – Don’t let that bother you. Marginal cost is a linear approximation of a curved function. For large values it gives a good approximation of the cost of producing the next item.
3.5 Derivatives of Trigonometric Functions slope Consider the function We could make a graph of the slope: Now we connect the dots! The resulting curve is a cosine curve.
3.5 Derivatives of Trigonometric Functions Find the derivative of cos x
3.5 Derivatives of Trigonometric Functions We can find the derivative of tangent x by using the quotient rule.
3.5 Derivatives of Trigonometric Functions Derivatives of the remaining trig functions can be determined the same way.
3.5 Derivatives of Trigonometric Functions Jerk A sudden change in acceleration DefinitionJerk Jerk is the derivative of acceleration. If a body’s position at time t is s(t), the body’s jerk at time t is
3.6 Chain Rule Consider a simple composite function:
3.6 Chain Rule If is the composite of and , then: Find: example: Chain Rule:
