# How do you find the derivative of #1/cosx#?

##### 1 Answer

It depends on what tools (theorems and definitions) you have to work with.

#### Explanation:

If we need to use the definition of derivative, we'll need the fundamental trigonometric limits:

along with

We'll also need the trigonometric identity:

**Here is the work.**

For

As always, the initial form of the limit of this difference quotient is indeterminate. We need to work with the difference quotient until we get a limit of determinate form.

We'll start by getting a single fractional expression.

# = (cosx-cos(x+h))/(hcos(x+h)cosx)#

Now expand

# = (cosx-cosxcos h+sinxsin h)/(hcos(x+h)cosx)#

Regroup so we can use thr fundamental trig limits.

# = (cosx(1-cos h)/h+sinxsin h/h)/(cos(x+h)cosx)#

Evaluate the limit by evaluating the individual limits

# = sinx/cos^2x = 1/cosx sinx/cosx = secx tanx#

**If you have the quotient rule**

Finish as above.

**If you have the chain rule**

Know that

And again finish as above.

**If you know the derivative of secant** , use