How do you use the chain rule to differentiate #log_(13)cscx#?

Answer 1

#(dy)/(dx)=-1/ln13*cotx#

Here ,

#y=log_13cscx#

Using Change of Base Formula:

#color(blue)(log_aX=log_k X/log_ka ,where,k " is the new base"#

So .

#y=log_ecscx/log_e13=1/ln13(lncscx)#

Let ,

#y=1/ln13lnu and u=cscx#
#(dy)/(du)=1/ln13*1/u and (du)/(dx)=-cscxcotx#

Using Chain Rule:

#color(red)((dy)/(dx)=(dy)/(du)*(du)/(dx#
#:.(dy)/(dx)=1/ln13*1/u(-cscxcotx)#
Subst. # u=cscx#
#:.(dy)/(dx)=1/ln13*1/(cscx)(-cscxcotx)#
#:.(dy)/(dx)=-1/ln13*cotx#
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Answer 2

Same as above answer, different method.

Alternatively...

Let #y = log_13(cscx)# #rArr y = -log_13(sinx)# #rArr sinx = 13^-y#
Using implicit differentiation: #d/dx(sinx) = d/dx(13^-y)# #rArr d/dx(sinx) = d/dy(e^(-y*ln13))*dy/dx# #:. cosx = -ln13*13^-y*dy/dx#
But #13^-y = sinx# #:. dy/dx = cosx/sinx -: -ln13# #= -cotx/ln13#

As above.

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Answer 3

To differentiate log base 13 of csc(x) with respect to x using the chain rule, follow these steps:

  1. Recognize that log base 13 of csc(x) can be expressed as log base 13 of (1/sin(x)).
  2. Apply the chain rule: differentiate the outer function (log base 13) and then multiply it by the derivative of the inner function (1/sin(x)).
  3. The derivative of log base 13 of u is (1/u)ln(13), where u = csc(x).
  4. The derivative of (1/sin(x)) is -csc(x)cot(x).
  5. Combine the results: (-csc(x)cot(x))(1/csc(x)ln(13)) = -cot(x)ln(13).
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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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