Deriving Trigonometric Functions

When finding derivatives of trigonometric expressions, always follow this three-step process: D power (derive the outer power), D trig (derive the trigonometric function), and D angle (derive the angle). The most important rule to remember is to never change the angle inside your trigonometric function during the first two steps.

Here are the basic trigonometric derivatives you need to memorize:

• sin x → cos x
• cos x → -sin x
• tan x → sec² x
• cot x → -csc² x
• sec x → sec x tan x
• csc x → -csc x cot x

💡 Success tip: Having the unit circle and Pythagorean identities (sin²x + cos²x = 1, 1 + tan²x = sec²x, 1 + cot²x = csc²x) memorized will make these problems much easier!

Let's see this in action with an example: Find y' of cos(5x)

1. D power: (1)cos(5x) - The power is just 1
2. D trig: -sin(5x) - Apply the derivative of cosine
3. D angle: -sin(5x)(5) - Multiply by the derivative of 5x Final answer: y' = -5sin(5x)

For more complex problems like sec²(6x):

1. D power: 2sec(6x)
2. D trig: 2sec(6x)tan(6x)
3. D angle: 2sec(6x)tan(6x)(6) Final answer: y' = 12sec(6x)tan(6x)

Even challenging problems like cot²$7x + 9x²$ follow the same pattern:

1. D power: 2cot$7x + 9x²$
2. D trig: -2csc²$7x + 9x²$
3. D angle: -2csc²7x + 9x²$$7 + 18x Final answer: y' = -$14 + 36x$csc²$7x + 9x²$