# How do you differentiate #f(x) = 4/sqrt(tan^2(1-x) # using the chain rule?

##### 1 Answer

See the explanation section below.

#### Explanation:

To avoid the quotient rule, let's rewrite

Now use the power and chain rules:

Finding

# = 4[(-1/2)(tan^2(1-x))^(-3/2)][2(tan(1-x))][sec^2(1-x)]d/dx(1-x)#

# = 4[(-1/2)(tan^2(1-x))^(-3/2)][2(tan(1-x))][sec^2(1-x)][-1]#

Simplifying algebraically, gets us,

# = (4tan(1-x)sec^2(1-x))/(tan^2(1-x))^(3/2)#

**Avoiding Error**

Remember that

but if we square first, we get

If you don't mind rewriting as a piecewise function, you could use:

Now differentiate each piece to get a piecewise derivative.