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How do you find the derivative of #y=e^(2x^3)#?

Answer 1

Ezra Adams

#(dy)/(dx)=6x^2e^(2x^3)#

Use the chain rule: #(dy)/(dx)=(dy)/(du)*(du)/(dx)#

#y=e^(2x^3),# let #u=2x^3#

#(dy)/(du)=e^u=e^(2x^3), (du)/(dx)=6x^2#

So #(dy)/(dx)=e^(2x^3)*6x^2=6x^2e^(2x^3)#

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Answer 2

Elijah Abram

To find the derivative of ( y = e^{2x^3} ), you can use the chain rule. The chain rule states that if ( y = f(g(x)) ), then ( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) ). In this case, ( f(u) = e^u ) and ( g(x) = 2x^3 ).

So, ( f'(u) = e^u ) and ( g'(x) = 6x^2 ).

Applying the chain rule:

[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) = e^{2x^3} \cdot 6x^2 = 6x^2 e^{2x^3} ]

So, the derivative of ( y = e^{2x^3} ) is ( 6x^2 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.