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Lesson 17 of 24
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# Derivatives of Trigonometric Functions

One of the most important types of motion in physics is simple harmonic motion, which is associated with such systems as an object with mass oscillating on a spring. Simple harmonic motion can be described by using either sine or cosine functions. In this section we expand our knowledge of derivative formulas to include derivatives of these and other trigonometric functions. We begin with the derivatives of the sine and cosine functions and then use them to obtain formulas for the derivatives of the remaining four trigonometric functions. Being able to calculate the derivatives of the sine and cosine functions will enable us to find the velocity and acceleration of simple harmonic motion.

# Derivatives of the Sine and Cosine Functions

We begin our exploration of the derivative for the sine function by using the formula to make a reasonable guess at its derivative. Recall that for a function .

Consequently, for values of  very close to 0, . We see that by using ,

By setting  and using a graphing utility, we can get a graph of an approximation to the derivative of  ((Figure)).

Upon inspection, the graph of  appears to be very close to the graph of the cosine function. Indeed, we will show that.

If we were to follow the same steps to approximate the derivative of the cosine function, we would find that.

### The Derivatives of  and

The derivative of the sine function is the cosine and the derivative of the cosine function is the negative sine.

## Proof

Because the proofs for  and  use similar techniques, we provide only the proof for . Before beginning, recall two important trigonometric limits we learned in Introduction to Limits: and .

The graphs of  and  are shown in (Figure).

We also recall the following trigonometric identity for the sine of the sum of two angles:.

Now that we have gathered all the necessary equations and identities, we proceed with the proof.

(Figure) shows the relationship between the graph of  and its derivative . Notice that at the points where  has a horizontal tangent, its derivative  takes on the value zero. We also see that where  is increasing,  and where  is decreasing, .

### Differentiating a Function Containing

Find the derivative of .

#### Solution

Using the product rule, we have

After simplifying, we obtain.

Find the derivative of

#### Hint

Don’t forget to use the product rule.

### Finding the Derivative of a Function Containing

Find the derivative of .

#### Solution

By applying the quotient rule, we have.

Simplifying, we obtain

Find the derivative of .

#### Hint

Use the quotient rule.

### An Application to Velocity

A particle moves along a coordinate axis in such a way that its position at time  is given by  for . At what times is the particle at rest?

#### Solution

To determine when the particle is at rest, set . Begin by finding . We obtain,

so we must solve for .

The solutions to this equation are  and . Thus the particle is at rest at times  and .

A particle moves along a coordinate axis. Its position at time  is given by  for . At what times is the particle at rest?

#### Hint

Use the previous example as a guide.

# Derivatives of Other Trigonometric Functions

Since the remaining four trigonometric functions may be expressed as quotients involving sine, cosine, or both, we can use the quotient rule to find formulas for their derivatives.

### The Derivative of the Tangent Function

Find the derivative of .

#### Solution

Start by expressing  as the quotient of  and :.

Now apply the quotient rule to obtain.

Simplifying, we obtain.

Recognizing that , by the Pythagorean Identity, we now have.

Finally, use the identity  to obtain.

Find the derivative of .

#### Hint

Rewrite  as  and use the quotient rule.

The derivatives of the remaining trigonometric functions may be obtained by using similar techniques. We provide these formulas in the following theorem.

### Derivatives of , and

The derivatives of the remaining trigonometric functions are as follows:

### Finding the Equation of a Tangent Line

Find the equation of a line tangent to the graph of  at .

#### Solution

To find the equation of the tangent line, we need a point and a slope at that point. To find the point, compute.

Thus the tangent line passes through the point . Next, find the slope by finding the derivative of  and evaluating it at : and .

Using the point-slope equation of the line, we obtain

or equivalently,.

### Finding the Derivative of Trigonometric Functions

Find the derivative of

#### Solution

To find this derivative, we must use both the sum rule and the product rule. Using the sum rule, we find.

In the first term, , and by applying the product rule to the second term we obtain.

Therefore, we have.

Find the derivative of .

#### Hint

Use the rule for differentiating a constant multiple and the rule for differentiating a difference of two functions.

Find the slope of the line tangent to the graph of  at .

#### Hint

Evaluate the derivative at .

# Higher-Order Derivatives

The higher-order derivatives of  and  follow a repeating pattern. By following the pattern, we can find any higher-order derivative of  and .

### Finding Higher-Order Derivatives of

Find the first four derivatives of .

#### Solution

Each step in the chain is straightforward:

#### Analysis

Once we recognize the pattern of derivatives, we can find any higher-order derivative by determining the step in the pattern to which it corresponds. For example, every fourth derivative of  equals , so

For , find .

#### Hint

See the previous example.

### Using the Pattern for Higher-Order Derivatives of

Find .

#### Solution

We can see right away that for the 74th derivative of , so.

For , find .

### An Application to Acceleration

A particle moves along a coordinate axis in such a way that its position at time  is given by . Find  and . Compare these values and decide whether the particle is speeding up or slowing down.

First find . Thus, . Next, find . Thus,  and we have . Since  and A block attached to a spring is moving vertically. Its position at time  is given by . Find  and . Compare these values and decide whether the block is speeding up or slowing down.

#### Solution

and . The block is speeding up.

### Key Concepts

• We can find the derivatives of  and  by using the definition of derivative and the limit formulas found earlier. The results are and .
• With these two formulas, we can determine the derivatives of all six basic trigonometric functions.

# Key Equations

• Derivative of sine function
• Derivative of cosine function
• Derivative of tangent function
• Derivative of cotangent function
• Derivative of secant function
• Derivative of cosecant function