Derivatives of Trig Functions

By now, you've seen how to differentiate polynomials, and perhaps a few other special functions (like logarithmic functions). In this lesson we'll learn some more derivatives, specifically the derivatives of the trigonometric functions: sine, cosine, tangent, secant, cosecant, and cotangent.

The characteristic trigonometric identity to recall in calculus is this:

It says that the derivative of sine is cosine, and the derivative of cosine is negative sine.

From these we may derive the rest of the derivatives, via the Quotient and Product rules. See if you can follow along as we derive them!

Derivative of Secant

Remember that the secant is the inverse of cosine -- it's 1/cos(x). Rewrite it as such, and find the derivative:

All we did was think of 1/cos(x) as cos(x) to the -1 power. Finally we simplified a bit by converting sin/cos to tan and the remaining 1/cos term to sec.

Derivative of Cosecant

Just like secant, except with sine instead of cosine. That's because cosecant is 1/sin!

Derivative of Tangent

For tangent, let's rewrite tangent as sin*sec. Remember that tangent is sin/cos, which is the same as sin*sec. It's often easier to differentiate a product, so we'll use that version and substitute the derivative of sec from above.

Derivative of Cotangent

To sum up: