Find the derivative using first principles? : #y=e^(2x)#

Answer 1

# dy/dx = 2e^(2x) #

Using the limit definition of the derivative:

# y=f(x) => dy/dx = lim_(h rarr 0) (f(x+h)-f(x))/h #
So if #y=e^(2x)#; then:
# dy/dx = lim_(h rarr 0) ( e^(2(x+h)) - e^(2(x)))/h # # \ \ \ \ \ \ = lim_(h rarr 0) ( e^(2x+2h) - e^(2x))/h # # \ \ \ \ \ \ = lim_(h rarr 0) ( e^(2x)e^(2h) - e^(2x))/h # # \ \ \ \ \ \ = lim_(h rarr 0) ( e^(2x)(e^(2h) - 1))/h # # \ \ \ \ \ \ = e^(2x) \ lim_(h rarr 0) ( e^(2h) - 1)/h # # \ \ \ \ \ \ = e^(2x) \ lim_(h rarr 0) ( 2(e^(2h) - 1))/(2h) # # \ \ \ \ \ \ = 2e^(2x) \ lim_(h rarr 0) ( e^(2h) - 1)/(2h) #
Then if we perform, a substitution, #alpha=2h# then clearly:
# h rarr 0 => alpha rarr 0 #

So we can write the derivative as:

# dy/dx = 2e^(2x) \ lim_(alpha rarr 0) ( e^(alpha) - 1)/(alpha) #
Now # lim_(alpha rarr 0) ( e^alpha - 1 ) / alpha = 1# is a standard calculus limit whose limit is unity, and so we find that:
# dy/dx = 2e^(2x) #
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Answer 2

To find the derivative of y = e^(2x) using first principles:

Let f(x) = e^(2x).

Then, using the definition of the derivative:

f'(x) = lim(h -> 0) [f(x + h) - f(x)] / h

Substitute f(x) into the formula:

f'(x) = lim(h -> 0) [(e^(2(x + h)) - e^(2x)) / h]

Expand using the properties of exponents:

f'(x) = lim(h -> 0) [(e^(2x + 2h) - e^(2x)) / h]

Apply the properties of exponents:

f'(x) = lim(h -> 0) [(e^(2x) * e^(2h) - e^(2x)) / h]

Factor out e^(2x):

f'(x) = lim(h -> 0) [e^(2x) * (e^(2h) - 1) / h]

As h approaches 0, e^(2h) approaches 1:

f'(x) = e^(2x) * lim(h -> 0) [(e^(2h) - 1) / h]

Now, recognize that the limit as h approaches 0 of [(e^(2h) - 1) / h] is the derivative of e^(2x) evaluated at x = 0, which is 2.

Therefore, the derivative of y = e^(2x) using first principles is:

f'(x) = 2e^(2x).

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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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