3.1 Derivative of a Function is called the derivative of at . We write: “The derivative of f with respect to x is …” There are many ways to write the derivative of

3.1 Derivative of a Function “f prime x” or “the derivative of f with respect to x” “y prime” “dee why dee ecks” “the derivative of y with respect to x” “dee eff dee ecks” “dee dee ecks uv eff uv ecks” “the derivative of f of x”

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 Note: does not mean ! (except when it is convenient to think of it as division.) does not mean ! (except when it is convenient to think of it as division.)

3.1 Derivative of a Function Note: does not mean times ! (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

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 To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: cusp corner vertical tangent discontinuity

3.2 Differentiability Most of the functions we study in calculus will be differentiable.

3.2 Differentiability There are two theorems on page 110: If f has a derivative at x = a, then f is continuous at x = a. Since a function must be continuous to have a derivative, if it has a derivative then it is continuous.

3.2 Differentiability Intermediate Value Theorem for Derivatives 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 .

3.3 Rules for Differentiation If the derivative of a function is its slope, then for a constant function, the derivative must be zero. example: The derivative of a constant is zero.

3.3 Rules for Differentiation We saw that if , . This is part of a pattern. examples: power rule

3.3 Rules for Differentiation Proof:

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 Find the horizontal tangents of: Horizontal tangents occur when slope = zero. 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

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 Higher Order Derivatives: is the first derivative of y with respect to x. is the second derivative. (y double prime) is the third derivative. We will learn later what these higher order derivatives are used for. is the fourth derivative.

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 :

3.3 Rules for Differentiation

3.3 Rules for Differentiation

3.4 Velocity and other Rates of Change Consider a graph of displacement (distance traveled) vs. time. Average velocity can be found by taking: time (hours) distance (miles) B A The speedometer in your car does not measure average velocity, but instantaneous velocity. (The velocity at one moment in time.)

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: Example: Free Fall Equation Speed is the absolute value of velocity.

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 time distance 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 = These definitions are true for any function. ( x does not have to represent time. )

3.4 Velocity and other Rates of Change For a circle: Instantaneous rate of change of the area with respect to the radius. For tree ring growth, if the change in area is constant then dr must get smaller as r gets larger.

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 Example 13: Suppose it costs: to produce x stoves. If you are currently producing 10 stoves, the 11th stove will cost approximately: The actual cost is: 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.

3.4 Velocity and other Rates of Change

3.5 Derivatives of Trigonometric Functions Consider the function slope 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 Proof

3.5 Derivatives of Trigonometric Functions = 0 = 1

3.5 Derivatives of Trigonometric Functions Find the derivative of cos x

3.5 Derivatives of Trigonometric Functions = 0 = 1

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 Definition Jerk 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

3.5 Derivatives of Trigonometric Functions

3.6 Chain Rule Consider a simple composite function:

3.6 Chain Rule Chain Rule: If is the composite of and , then: Find: example: Find:

3.6 Chain Rule

3.6 Chain Rule Here is a faster way to find the derivative: Differentiate the outside function... …then the inside function

3.6 Chain Rule The chain rule can be used more than once. (That’s what makes the “chain” in the “chain rule”!)

3.6 Chain Rule Derivative formulas include the chain rule! etcetera…

3.6 Chain Rule Find

3.6 Chain Rule The chain rule enables us to find the slope of parametrically defined curves: The slope of a parametrized curve is given by:

3.6 Chain Rule Example: These are the equations for an ellipse.

3.7 Implicit Differentiation This is not a function, but it would still be nice to be able to find the slope. Do the same thing to both sides. Note use of chain rule.

3.7 Implicit Differentiation This can’t be solved for y. This technique is called implicit differentiation. 1 Differentiate both sides w.r.t. x. 2 Solve for .

3.7 Implicit Differentiation Implicit Differentiation Process Differentiate both sides of the equation with respect to x. Collect the terms with dy/dx on one side of the equation. Factor out dy/dx . Solve for dy/dx .

3.7 Implicit Differentiation Find the equations of the lines tangent and normal to the curve at . Note product rule.

3.7 Implicit Differentiation Find the equations of the lines tangent and normal to the curve at . normal: tangent:

3.7 Implicit Differentiation

3.7 Implicit Differentiation Find if . Substitute back into the equation.

3.7 Implicit Differentiation Rational Powers of Differentiable Functions Power Rule for Rational Powers of x If n is any rational number, then

3.7 Implicit Differentiation Proof: Let p and q be integers with q > 0. Raise both sides to the q power Differentiate with respect to x Solve for dy/dx

3.7 Implicit Differentiation Substitute for y Remove parenthesis Subtract exponents

3.8 Derivatives of Inverse Trigonometric Functions Slopes are reciprocals. Because x and y are reversed to find the reciprocal function, the following pattern always holds: The derivative of Derivative Formula for Inverses: evaluated at is equal to the reciprocal of the derivative of evaluated at .

3.8 Derivatives of Inverse Trigonometric Functions We can use implicit differentiation to find:

3.8 Derivatives of Inverse Trigonometric Functions We can use implicit differentiation to find: But so is positive.

3.8 Derivatives of Inverse Trigonometric Functions

3.8 Derivatives of Inverse Trigonometric Functions Find

3.8 Derivatives of Inverse Trigonometric Functions Find

3.8 Derivatives of Inverse Trigonometric Functions

3.8 Derivatives of Inverse Trigonometric Functions Your calculator contains all six inverse trig functions. However it is occasionally still useful to know the following:

3.8 Derivatives of Inverse Trigonometric Functions Find

3.9 Derivatives of Exponential and Logarithmic Functions Look at the graph of If we assume this to be true, then: The slope at x = 0 appears to be 1. definition of derivative

3.9 Derivatives of Exponential and Logarithmic Functions Now we attempt to find a general formula for the derivative of using the definition. This is the slope at x = 0, which we have assumed to be 1.

3.9 Derivatives of Exponential and Logarithmic Functions is its own derivative! If we incorporate the chain rule: We can now use this formula to find the derivative of

3.9 Derivatives of Exponential and Logarithmic Functions Incorporating the chain rule:

3.9 Derivatives of Exponential and Logarithmic Functions So far today we have: Now it is relatively easy to find the derivative of .

3.9 Derivatives of Exponential and Logarithmic Functions

3.9 Derivatives of Exponential and Logarithmic Functions To find the derivative of a common log function, you could just use the change of base rule for logs: The formula for the derivative of a log of any base other than e is:

3.9 Derivatives of Exponential and Logarithmic Functions

3.9 Derivatives of Exponential and Logarithmic Functions Find y’

3.9 Derivatives of Exponential and Logarithmic Functions Logarithmic differentiation Used when the variable is in the base and the exponent y = xx ln y = ln xx ln y = x ln x