How do I find the derivative of #3e^(-12t) #?

Answer 1

You can use the chain rule.

#(3e^(-12t))'=-36*e^(-12t)#

The 3 is a constant, it can be kept out:

#(3e^(-12t))'=3(e^(-12t))'#

It's a mixed function. The outer function is the exponential, and the inner is a polynomial (sort of):

#3(e^(-12t))'=3*e^(-12t)*(-12t)'=#
#=3*e^(-12t)*(-12)=-36*e^(-12t)#

Deriving:

If the exponent was a simple variable and not a function, we would simply differentiate #e^x#. However, the exponent is a function and should be transformed. Let #(3e^(-12t))=y# and #-12t=z#, then the derivative is:
#(dy)/dt=(dy)/dt*(dz)/dz=(dy)/dz*(dz)/dt#
Which means you differentiate #e^(-12t)# as if it were #e^x# (unchanged), then you differentiate #z# which is #-12t# and finally you multiply them.
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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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