How do you differentiate #2^(2x)#?

Answer 1

Unless told that I must use the chain rule, I would not. But see below.

#2^(2x) = (2^2)^x = 4^x#
#d/dx(a^x) = a^x lna# #" "# (A proof of this uses the chain rule.)
So #d/dx(2^(2x)) = d/dx(4^x) = 4^x ln4#
Proof of #d/dx(a^x) = a^x lna#
#(a^x) = (e^ lna)^x = e^(xlna) #
#d/dx(a^x) = d/dx(e^(xlna)) = e^(xlna) d/dx(xlna)#
Note that #lna# is a constant, so that #d/dx(xlna) = lna#.

We continue:

# = e^(xlna) d/dx(xlna) = e^(xlna) lna #
Finally, recall that #e^(xlna) = a^x# (see above), so we finish
# = e^(xlna) lna = a^x lna #
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Answer 2

To differentiate (2^{2x}), you can use the chain rule. First, take the derivative of the exponent, which is (2x), then multiply by the derivative of the base, which is (2^x) times the natural logarithm of 2 ((2^x \cdot \ln(2))). So the differentiation of (2^{2x}) is (2^{2x} \cdot \ln(2) \cdot 2). Thus, the derivative is (4^x \cdot \ln(2)).

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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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