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What is the derivative of #y=3x^2e^(5x)# ?

Answer 1

This is a product of the function #3x^2# and the function #e^(5x)#, which is itself the composite of the functions given by #e^x# and by #5x#.

Thus we will need the product rule to the effect that:

#(3x^2 e^(5x))'=(3x^2)'e^(5x)+3x^2(e^(5x))'#

As well as the fact that:

#(3x^2)'=6x#

Using these basic differentiation rules, and the Chain Rule combined with #(e^x)'=e^x#, we can calculate that:

#(e^(5x))'=e^(5x)\times (5x)'=5e^(5x)#.

As a result:

#(3x^2 e^(5x))'=6xe^(5x)+15x^2e^(5x)#.

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Answer 2

Aurora Alvarado

To find the derivative of ( y = 3x^2 e^{5x} ), you can use the product rule and the chain rule. The derivative is:

[ \frac{dy}{dx} = 6xe^{5x} + 3x^2 \cdot 5e^{5x} ]

[ \frac{dy}{dx} = 6xe^{5x} + 15x^2e^{5x} ]

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Answer 3

Elijah Abram

To find the derivative of y = 3x^2e^(5x), you can use the product rule and the chain rule. Here's how:

Let u = 3x^2 and v = e^(5x).

Then, using the product rule (d(uv)/dx = u'v + uv'), we have:

u' = 6x v' = 5e^(5x)

Now, applying the product rule:

y' = u'v + uv' = (6x)(e^(5x)) + (3x^2)(5e^(5x)) = 6xe^(5x) + 15x^2e^(5x)

So, the derivative of y = 3x^2e^(5x) is y' = 6xe^(5x) + 15x^2e^(5x).

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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.