Find the derivative of
𝑦
=
𝑥
2
𝑒
4
𝑥

Question

Find the derivative of 
𝑦
=
𝑥
2
𝑒
4
𝑥

BlackTom AI · Accepted Answer

We start with the function y = x^2 e^{4x} and need its derivative.

Option structure is based on the product rule, since y is a product of x^2 and e^{4x}.

Applying the product rule: if u = x^2 and v = e^{4x}, then y' = u'v + uv'.
- u' = d/dx(x^2) = 2x.
- v' = d/dx(e^{4x}) = e^{4x} * 4, by the chain rule.

Thus y' = (2x) e^{4x} + x^2 (e^{4x}) * 4.

This can be written as y' = 2x e^{4x} + 4x^2 e^{4x} = e^{4x}(2x + 4x^2).

In the given option notation, the term (2x) e^{4x} matches the first part, and the second part x^2 e^{4x} (4) is the same as x^2 e^{4x} * 4, representing the derivative of e^{4x} scaled by x^2.

For completeness, you can factor out e^{4x} to see the common factor: y' = e^{4x}(2x + 4x^2) = 2x e^{4x} (1 + 2x).

Note about the options: there are no alternative choices provided to compare against. If other options existed, you would evaluate them by applying the product rule correctly (and the chain rule for the exponential part) and simplify to see which matches the expression above. In any case, the derivative computation shown here follows the standard rules of differentiation for a product of a polynomial and an exponential function.

ST-MATH-1031-770-49505-202503 Test 2

Find the derivative of 𝑦 = 𝑥 2 𝑒 4 𝑥

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