In this explainer, we will learn how to find the derivatives of trigonometric functions and how to apply the differentiation rules on them.
Trigonometric functions and their derivatives have several real-world applications in various fields such as physics, engineering, architecture, robotics, music theory, and navigation, to name a few. In physics, it can be used in projectile motion, to model the mechanics of electromagnetic waves, analyzing alternating and direct currents, and finding the trajectory of a mass around a massive body under the force of gravity.
In this explainer, we will particularly be interested in the derivatives of the sine, cosine, and tangent functions. We will first determine the derivative of the standard trigonometric functions, starting with , from first principles, and then use that result to also determine the derivatives of and using the chain and quotient rule.
Letβs first recall the definition of the derivative.
Definition: The Derivative
The derivative of a function is defined as at the points where the limit exists.
Substituting into the definition of the derivative, we have
To simplify the expression within the limit, we will make use of trigonometric identities. In particular, we can use the sum identity,
Using this identity, we can expand the and rewrite the resulting expression using the sum and product rules of limits to obtain
Since both and are independent of , their limits will simply be and respectively. Hence, the derivative becomes
The other two limits are not as trivial but are standard results:
- ;
- .
We can also determine the second limit from the first by applying trigonometric identities. In order to see this, we begin by multiplying the numerator and denominator by , which yields
Using the Pythagorean identity for sine and cosine, , we can rewrite the numerator in terms of sine and then apply the multiplicative law and take the resulting limits:
Substituting the value of these two limits into , we obtain the derivative of as follows:
We can also determine the derivative of from first principles in a similar way; however, it is better to determine the result from the cofunction identity and the chain rule with .
Rule: The Chain Rule
Given two differentiable functions and , the derivative of their composition is given by
This can be written succinctly using prime notation as follows:
We can rewrite the function as with , and the derivatives are
Thus, using the chain rule, the derivative of is
We can rewrite the final result using the other cofunction identity, , in reverse:
Now, letβs consider higher derivatives of and that will form a cyclic pattern in the order of the derivatives. For , the first four derivatives are
Thus, the fourth derivative gives you back the original function, and higher derivatives of sine form a repeating cyclic pattern with period 4. This can be depicted in the following diagram.
The general derivatives for can be written as
Similarly, for , the first four derivatives are
Thus, the fourth derivative gives you back the original function and higher derivatives of cosine form a repeating cyclic pattern with period 4, similar to the sine function. This can be depicted in the following diagram.
The general derivatives for can be written as
Now, letβs consider an example where we determine a particular higher-order derivative of sine by using the cyclic pattern.
Example 1: Consecutive Derivatives of Sine
Find the thirty-third derivative of .
Answer
In this example, we want to determine the thirty-third derivative of the sine function.
For , the first four derivatives are
Thus, the fourth derivative gives you back the original function and this cyclic pattern will repeat for higher-order derivatives with period 4. The general rule for the derivative with is
Since , we have . Hence, the thirty-third derivative is equivalent to the first derivative of and we have
The derivative of can be found by using the quotient rule; by using the trigonometric identity that , we can express as
To find the derivative of , we can use the quotient rule.
Rule: The Quotient Rule
Given two differentiable functions and , the derivative of their quotient is given by
This can be written succinctly using prime notation as follows:
Setting and , we begin by differentiating and to find expressions for and from previously established results:
Substituting these expressions into the quotient rule, we have
Using the Pythagorean identity for sine and cosine, we can simplify the numerator. Hence,
In general, we can establish a rule for derivatives of the function using the chain rule; we can rewrite this as , where , and using the derivative becomes
Similarly, for , using the chain rule by rewriting the function as , where , we obtain the general derivative
The derivative of can also be found using the chain rule by rewriting the function as , where , or using the quotient rule with the derivatives of the general sine and cosine functions to obtain
Now, we summarize the general derivatives of trigonometric functions as established.
Standard Result: Derivatives of Trigonometric Functions
The general derivatives of the sine, cosine, and tangent functions are as follows: for .
Using these standard results, we can find the derivatives of a large class of functions containing sums of trigonometric functions using the linearity of the derivative:
Other functions that have compositions, products, or quotients of trigonometric functions use the chain rule, the product rule, the quotient rule, or any combination of them.
In the next example, we will differentiate a function containing a sum of polynomial and trigonometric functions using the linearity of the derivative.
Example 2: Differentiating a Combination of Polynomial and Trigonometric Functions
If , find .
Answer
In this example, we want to differentiate a function that has a combination of a polynomial and trigonometric functions.
The power rule and rules for differentiating the sine are
Applying these rules, we can calculate the first derivative of as
In the next example, we will use the chain rule to find the derivative of a function composed of a quadratic function and the sum of sine and cosine functions.
Example 3: Differentiating Trigonometric Functions
If , find .
Answer
In this example, we want to determine the derivative of a composite function using the chain rule and the general rule for differentiating the sine and cosine functions.
Recall that the chain rule for a composite function is given by for differentiable functions and . The power rule and the rules for differentiating the sine and cosine are
The given function is a composite, where and . We begin by differentiating with respect to using the power rule and with respect to using the rules for differentiating the sine and cosine:
Finally, substituting these expressions back into the chain rule where , we find the derivative of with respect to as
We can simplify this further by applying the double angle trigonometric identity: where ; we can simplify the derivative to obtain
We note that we could have also obtained this answer by first distributing the brackets in the given function, applying a trigonometric identity, and then differentiating the resulting expression.
Now, letβs consider an example where we will use the chain rule to find the derivative of a function composed of a square function and a tangent function.
Example 4: Finding the Derivative of a Trigonometric Function Using the Chain Rule
Find the derivative of the function .
Answer
In this example, we want to determine the derivative of a composite function using the chain rule and the general rule for differentiating the tangent functions.
Recall that the chain rule for a composite function is given by for differentiable functions and . The power rule and the rules for differentiating the tangent function are
The given function is a composite function, where , where . We begin by differentiating with respect to using the power rule and with respect to using the rules for differentiating the tangent function:
Finally, substituting these expressions back into the chain rule where , we find the derivative of with respect to as follows:
We can also differentiate functions that contain trigonometric functions as products using the product rule.
Rule: The Product Rule
Given two differentiable functions and , the derivative of their product is given by
This can be written succinctly using prime notation as follows:
In the next example, we will find the derivative of a function that contains a product of a power and a sine function.
Example 5: Differentiating Functions Involving Trigonometric Ratios Using the Product Rule
If , determine .
Answer
In this example, we want to find the derivative of a product function that contains a trigonometric function using the product rule.
Recall that the product rule for the derivative of a product function is for differentiable functions and . The power rule and the rules for differentiating the sine are
The given function is a product of two functions, where , where and . We begin by differentiating and with respect to using the power rule and the rule for differentiating the sine as
Finally, substituting these expressions back into the product rule, we find the derivative of with respect to as
We can also find the derivative of trigonometric functions evaluated at a particular point, which is the same as the gradient of the tangent line to the curve at that point.
Now, letβs consider an example where we find the derivative of a function containing the product of a sine and tangent function using the product rule and evaluate the derivative at the given point.
Example 6: Differentiating Trigonometric Functions Using the Product Rule
Given , find at .
Answer
In this example, we want to find the derivative of a product function that contains a trigonometric function using the product rule and then evaluate this at the given point.
Recall that the product rule for the derivative of a product function is for differentiable functions and . The rules for differentiating the sine and tangent functions are
The given function is a product of two functions, where , where and . We begin by differentiating and with respect to using the product rule and the rule for differentiating the sine and tangent as follows:
Substituting these expressions back into the product rule, we find the derivative of with respect to as follows:
Finally, substituting , we find the derivative at this point:
We can also differentiate functions that contain trigonometric functions as quotients, either in the numerator, denominator, or both, using the quotient rule, similar to how we found the derivative of the tangent function.
Now, letβs consider an example wherewe determine the derivative of a quotient of a linear function, in the numerator, and a tangent function, in the denominator.
Example 7: Finding the First Derivative of the Quotient of Trigonometric and Linear Functions Using the Quotient Rule
Differentiate .
Answer
In this example, we want to find the derivative of a quotient function that contains a trigonometric function using the quotient rule.
Recall that the quotient rule for the derivative of a quotient function is for differentiable functions and . The power rule and rules for differentiating the tangent function are
The given function is a quotient of two functions, where , where and . We begin by differentiating and with respect to using the power rule and the rule for differentiating the tangent function as follows:
Finally, substituting these expressions back into the quotient rule, we find the derivative of with respect to :
In the next example, we will find the derivative of a function containing the quotient of a cosine function in the numerator and the sine function in the denominator, using the quotient rule, and evaluate the derivative at the given point.
Example 8: Finding the First Derivative of the Quotient of Trigonometric Functions Using the Quotient Rule at a Point
Given that , determine at .
Answer
In this example, we want to find the derivative of a quotient function that contains a trigonometric function using the quotient rule and then evaluate this at the given point.
Recall that the quotient rule for the derivative of a quotient function is for differentiable functions and . The rules for differentiating the sine and cosine are
The given function is a quotient of two functions, where , where and . We begin by differentiating and with respect to using the power rule and the rule for differentiating the tangent function as follows:
Substituting these expressions back into the quotient rule, we find the derivative of with respect to :
Using the Pythagorean identity , we can simplify the expression as
Finally, substituting , we find the derivative at this point as
In the final example, we will determine the derivative of a function that contains a quotient with a product of a linear function and sine function in the numerator and another linear function in the denominator. We will make use of both the product rule and the quotient rule.
Example 9: Differentiating a Combination of Linear and Trigonometric Functions Using the Quotient Rule
Differentiate .
Answer
In this example, we want to find the derivative of a quotient function that contains a trigonometric function and product function in the numerator, using the product and quotient rule.
Recall that the quotient rule for the derivative of a quotient function is for differentiable functions and . Also, the product rule for the derivative of a product function is for differentiable functions and . The power rule and the rules for differentiating the sine are
The given function is a quotient of two functions , where and . We begin by differentiating and with respect to .
We first note that the numerator is also a product of two functions , where and and thus the derivative of can be found by applying the product rule with the derivatives of and using the power rule and the rule for differentiating the sine as follows: and thus the derivative of with respect to is
The derivative of with respect to can be found by applying the power rule:
Substituting these expressions back into the quotient rule, we find the derivative of with respect to :
Letβs now summarize a few key points from the explainer.
Key Points
- We can determine the standard results for the derivatives of trigonometric functions from first principles using the definition of the derivative. It is sufficient to do this for the sine function and then find the results of the cosine and tangent function using trigonometric identities along with the chain and quotient rule. In particular, we find
- The higher derivatives of sine and cosine have a cyclic pattern that repeats with period 4; the fourth derivative of either function gives you back the original function: This implies the following general rules for the derivative with :
- The general derivatives of the sine, cosine, and tangent functions can be found from the chain rule and are as follows: for .
- Using these standard results, we can find the derivatives of a large class of functions containing trigonometric functions using the linearity of the derivative, the chain rule, the product rule, and the quotient rule.
