How do you differentiate #tan(3x^2) - csc ( ln(4x) )^2#?

Answer 1

#6xsec^2(3x^2)+(2*csc^2(ln(4x))*cot(ln(4x)))/x#

Wow, this looks tough. Let's break it down piece by piece.

First, let's take a look at the #tan(3x^2)# term. We apply the chain rule when we differentiate this term.
We first differentiate this term with respect to the inner function #3x^2# and then multiply the result by the derivative of #3x^2# with respect to #x# (By the chain rule).
#(tan(3x^2))^' = sec^2(3x^2)*(6x)=6xsec^2(3x^2)#
Not too bad! :) (Remember: #d/dx(tanx)=sec^2(x)#)
Now for the other (very scary) term, #-csc^2(ln(4x))#.
Wait a minute, that's not too bad. There are just four functions nested in each other; the #4x# in the #ln()#, the #ln()# in the #csc()# and finally the #csc()# in the #()^2#.

This means that we will need to apply the chain rule three times.

First we see that,

#(-csc^2(ln(4x)))^'=[-2csc(ln(4x))]*[csc(ln(4x))]^'#

Now,

# [csc(ln(4x))]^' = [-csc(ln(4x))*cot(ln(4x))][ln(4x)]^'#

and

#[ln(4x)]^' = [1/(4x)]*[4x]^' = 4(1/(4x))= 1/x#

Putting everything together we have

#-csc^2(ln(4x)))^'=([-2csc(ln(4x))]*[-csc(ln(4x))*cot(ln(4x))])/x# #=(2*csc^2(ln(4x))*cot(ln(4x)))/x#
So finally, adding on #6xsec^2(3x^2)#, we have the derivative of the given expression as
#6xsec^2(3x^2)+(2*csc^2(ln(4x))*cot(ln(4x)))/x#

Woohoo! :)

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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