How do you find the derivative of #1/cosx#?
It depends on what tools (theorems and definitions) you have to work with.
If we need to use the definition of derivative, we'll need the fundamental trigonometric limits:
We'll also need the trigonometric identity:
Here is the work.
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)#
# = (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
And again finish as above.
If you know the derivative of secant , use