2.3 The Product and Quotient Rules and Higher Order Derivatives

After this lesson, you should be able to: Find the derivative of a function using the Product Rule Find the derivative of a function using the Quotient Rule Find the derivative of a trigonometric function Find a higher-order derivative of a function

Theorem 2.7 The Product Rule The product of two differentiable functions f and g is itself differentiable. The derivative is where Example Find dy/dx

Proof of Theorem 2.7 Since f (x) and g(x) are differentiable, so g(x) is continuous. and exist. Therefore exist. and exist. Therefore exist.

Proof of Theorem 2.7

The Product Rule Example Example Find dy / dx :

The Product Rule Example Example Find f ’(x)

The Product Rule Example Example Find y’ Ans:

The Quotient Rule The quotient of two differentiable functions f and g is itself differentiable at all values of x for which g(x) ≠ 0. The derivative is: where Example Find y ’

Proof of Theorem 2.8, The Quotient Rule Since f (x) and g(x) are differentiable, so g(x) is continuous. and exist. Therefore exist. and exist. Therefore exist.

Proof of Theorem 2.8, The Quotient Rule

Proof of Theorem 2.8, The Quotient Rule

The Quotient Rule Example Example Find f ’(x):

Equation of a Horizontal Tangent Line Find the equations of the horizontal tangent lines for Since we are asked to find horizontal tangent lines, we know the slopes of these lines are 0. So, set the 1st derivative equal to 0, then solve for x. This gives us the x-values of points on the graph where there are horizontal tangent lines. Then, we can find the points and write the equations. Therefore, the points where there are horizontal tangent lines are Since the tangent lines are horizontal, the equations are

The Quotient Rule Example

More Quotient Rule Examples Sometimes, you need rewrite or simplify the function before you try to take the derivative(s). Example Rewrite Then

More Quotient Rule Examples In Section 2.2, the Power Rule was provided only for the case where the exponent n is a positive integer greater then 1. By using the Quotient Rule, we can generalize the Power Rule to n is an integer. If n is a negative integer, there exists a positive integer k such that n = –k So, by the quotient rule

The Derivative of Tangent

The Derivative of Secant

The Derivatives of Cosecant and Cotangent

Equation of the Tangent Line Example Given Write the equation of the tangent line at We can find the derivative of f (x) by using the rules. The equation of the tangent line at is

Note Because of the trigonometric identities, the derivative of a trigonometric function may have many forms. This presents a challenge when you are trying to match your answers to those given ones.

Higher Order Derivatives First Derivative Ex: Second Derivative Ex: Third Derivative Ex: Fourth Derivative Ex: n-th Derivative Ex:

Higher Order Derivatives of sin(x)

Homework Section 2.3 page 124 #1-23odd, 27, 41, 43, 51, 89, 91, 101, 103