Derived From Trigonometric Functions

Q: 📐 Do I need to memorize all six trigonometric derivative formulas?

While ideally yes, you could focus on just three core ones: <ul> <li class="font-claude-response-body break-words whitespace-normal" dir="ltr">sine,</li> <li class="font-claude-response-body break-words whitespace-normal" dir="ltr">cosine, and</li> <li class="font-claude-response-body break-words whitespace-normal" dir="ltr">tangent.</li> </ul> These are the most commonly used in calculus courses. The other three (cotangent, secant, cosecant) are less frequent but follow similar patterns, so once you master the main three using the chain rule, the others become easier to understand.

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Formulas for Deriving Trigonometric Functions

Derivative of the Sine Function

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Derivative of the Cosine Function

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Derivative of the Tangent Function

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Derivative of the Cotangent Function

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Derivative of the Secant Function

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Derivative of the Cosecant Function

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Examples of Trigonometric Function Derivative Exercises

Derive the following functions.

Remember: Always derive the argument of the trigonometric function and multiply it by the derivative of the function.

1 $\text{[math]}$

a. First we set $\text{[math]}$

b. We calculate the derivative of $\text{[math]}$

$\text{[math]}$

c. We substitute into the sine derivative formula

$\text{[math]}$

d. Reordering, we have

$\text{[math]}$

2 $\text{[math]}$

a. First we set $\text{[math]}$

b. We calculate the derivative of $\text{[math]}$

$\text{[math]}$

c. We substitute into the sine derivative formula

$\text{[math]}$

d. Reordering, we have

$\text{[math]}$

3 $\text{[math]}$

a. First we set $\text{[math]}$

b. We calculate the derivative of $\text{[math]}$

$\text{[math]}$

c. We substitute into the power function derivative formula

$\text{[math]}$

4 $\text{[math]}$

a. First we set $\text{[math]}$

b. We calculate the derivative of $\text{[math]}$

$\text{[math]}$

c. We substitute into the cosine derivative formula

$\text{[math]}$

d. Reordering, we have

$\text{[math]}$

5 $\text{[math]}$

a. First we set $\text{[math]}$

b. We calculate the derivative of $\text{[math]}$

$\text{[math]}$

c. We substitute into the cosine derivative formula

$\text{[math]}$

d. Reordering, we have

$\text{[math]}$

6 $\text{[math]}$

a. First we set $\text{[math]}$

b. We calculate the derivative of $\text{[math]}$

$\text{[math]}$

c. We substitute into the power function derivative formula

$\text{[math]}$

d. Reordering, we have

$\text{[math]}$

7 $\text{[math]}$

a. First we set $\text{[math]}$

b. We calculate the derivative of $\text{[math]}$

$\text{[math]}$

c. We substitute into the tangent derivative formula

$\text{[math]}$

d. Reordering, we have

$\text{[math]}$

8 $\text{[math]}$

a. First we set $\text{[math]}$

b. We calculate the derivative of $\text{[math]}$

$\text{[math]}$

c. We substitute into the cotangent derivative formula

$\text{[math]}$

d. Reordering, we have

$\text{[math]}$

9 $\text{[math]}$

a. First we set $\text{[math]}$

b. We calculate the derivative of $\text{[math]}$

$\text{[math]}$

c. We substitute into the power function derivative formula

$\text{[math]}$

d. Reordering, we have

$\text{[math]}$

10 $\text{[math]}$

a. First we set $\text{[math]}$

b. We calculate the derivative of $\text{[math]}$

$\text{[math]}$

c. We substitute into the secant derivative formula

$\text{[math]}$

d. Reordering, we have

$\text{[math]}$

11 $\text{[math]}$

a. First we set $\text{[math]}$

b. We calculate the derivative of $\text{[math]}$

$\text{[math]}$

c. We substitute into the cosecant derivative formula

$\text{[math]}$

d. Reordering, we have

$\text{[math]}$

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Frequently asked questions

📐 Do I need to memorize all six trigonometric derivative formulas?

While ideally yes, you could focus on just three core ones:

sine,
cosine, and
tangent.

These are the most commonly used in calculus courses. The other three (cotangent, secant, cosecant) are less frequent but follow similar patterns, so once you master the main three using the chain rule, the others become easier to understand.

🔗 Why do I always need to apply the chain rule when deriving trigonometric functions?

The chain rule applies whenever you have a trigonometric function with something other than just "x" inside it. For example, if you're deriving sine of 4x, you can't ignore that 4; you must multiply by its derivative. Even with simple arguments like 4x or more complex ones like polynomials, always identify what's inside the trig function, derive that part separately, and multiply.

Chapters

Formulas for Deriving Trigonometric Functions
Examples of Trigonometric Function Derivative Exercises

An Expert Guide on Solving a Derivative of Trigonometric Functions