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How do you differentiate #y=e^((2x)/3)#?

Answer 1

Scarlett Allison

#dy/dx=2/3e^((2x)/3)#

#color(orange)"Reminder"#

#d/dx(e^x)=e^x#

and using the #color(blue)"chain rule"#

#color(red)(|bar(ul(color(white)(a/a)color(black)(d/dx(e^(g(x)))=e^(g(x))g'(x))color(white)(a/a)|)))#

#rArrdy/dx=e^((2x)/3).d/dx(2/3x)=2/3e^((2x)/3)#

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Answer 2

Christopher Agresta

To differentiate ( y = e^{(2x/3)} ), you would use the chain rule, which states that if ( y = f(g(x)) ), then ( \frac{{dy}}{{dx}} = f'(g(x)) \cdot g'(x) ).

First, find the derivative of the outer function ( e^u ), which is ( e^u ) itself. Then, find the derivative of the inner function ( u = \frac{2x}{3} ), which is ( \frac{2}{3} ).

So, the derivative of ( y = e^{(2x/3)} ) is ( \frac{2}{3} \cdot e^{(2x/3)} ).

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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.