2.4

Derivatives of Trigonometric Functions

Example 1

Differentiate y = x2 sin x.

Solution:Using the Product Rule

Example 2

An object at the end of a vertical spring is stretched 4 cm beyond its rest position and released at time t = 0 (note that the downward direction is positive.)

Its position at time t is

s = f (t) = 4 cos t

Find the velocity and acceleration

at time t and use them to analyze

the motion of the object.

Example 2 – Solution

• The velocity and acceleration are

Example 2 – Solution

The object oscillates from the lowest point (s = 4 cm) to the highest point (s = –4 cm). The period of the oscillation is 2, which is the period of cos t.

cont’d

Example 3 – Solution

• The speed is | v | = 4 | sin t |, which is greatest when | sin t | = 1, that is, when cos t = 0.

• So the object moves fastest as it passes through its equilibrium position (s = 0). Its speed is 0 when sin t = 0, that is, at the high and low points.

• The acceleration a = –4 cos t = 0 when s = 0. It has greatest magnitude at the high and low points.

cont’d

2.5

The Chain Rule

The Chain Rule

Example

• Find F '(x) if F (x) = .

Solution: (using the first definition)• F (x) = (f g)(x) = f (g(x)) where f (u) = and g (x) = x2 + 1.

• Since and g(x) = 2x

we have F (x) = f (g (x)) g (x)

Solution: (using the second definition)

• let u = x2 + 1 and y = , then

The Chain Rule for powers:

Example• Differentiate y = (x3 – 1)100.

• Solution:Taking u = g(x) = x3 – 1 and n = 100

= (x3 – 1)100

= 100(x3 – 1)99 (x3 – 1)

= 100(x3 – 1)99 3x2

= 300x2(x3 – 1)99

2.6

Implicit Differentiation

Implicit Differentiation

So far we worked with functions where one variable is expressed in terms of another variable—for example:y = or y = x sin x (in general: y = f (x). )

Some functions, however, are defined implicitly by a relation between x and y, examples:

x2 + y2 = 25x3 + y3 = 6xy

We say that f is a function defined implicitly - For example Equation 2 above means: x3 + [f (x)]3 = 6x f (x)

Example 1

For x3 + y3 = 6xy find:

Find the equation of the tangent line at (3,2)

Example 2

Answer:

Practice problem:Find the slope of the curve at (4,4)

Answer: