Differentiate

𝑒
3
𝑥
2
+
5
𝑥

Question

Differentiate

𝑒
3
𝑥
2
+
5
𝑥

BlackTom AI · Accepted Answer

We start by restating the problem to ensure clarity: Differentiate the function e^{3x^2} + 5x.

Now, examine the given proposed answer: (6x + 5) e^{3x^2} + 5x. This expression is not consistent with standard differentiation rules for the following reason: the derivative of a sum is the sum of the derivatives, so we must differentiate each term separately.

First term: differentiate e^{3x^2}.
- Recognize that e^{g(x)} differentiates to e^{g(x)} * g'(x).
- Here, g(x) = 3x^2, so g'(x) = 6x.
- Therefore, d/dx [e^{3x^2}] = e^{3x^2} * 6x = 6x e^{3x^2}.

Second term: differentiate 5x.
- The derivative of a constant multiple of x is the constant itself, so d/dx [5x] = 5.

Combine the results: derivative of e^{3x^2} + 5x is 6x e^{3x^2} + 5.

Now compare with the proposed answer: (6x + 5) e^{3x^2} + 5x.
- This expression incorrectly places the 5 inside the exponential factor and duplicates x terms in a way that does not match the linearity of differentiation.
- Specifically, 6x e^{3x^2} should be a product of 6x and e^{3x^2}, not (6x + 5) times e^{3x^2}.
- Also, there should be no extra +5x term in the derivative since the derivative of 5x is 5, not 5x.

If we rewrite the correct derivative clearly, it is: 6x e^{3x^2} + 5.

Interpreting potential confusion: some students may misapply the product rule or fail to apply the chain rule properly. The chain rule requires multiplying the derivative of the outer function e^{u} by the derivative of the inner function u = 3x^2, yielding 6x e^{3x^2}. The linear term 5x contributes a constant 5 to the derivative, not an additional x-term.

To summarize the key points:
- Correct derivative of e^{3x^2} is 6x e^{3x^2} (via chain rule).
- Correct derivative of 5x is 5.
- Correct combined derivative is 6x e^{3x^2} + 5.

Therefore, the proposed answer is incorrect due to misapplication of the product/chain rules and incorrect distribution of terms.

Differentiate 𝑒 3 𝑥 2 + 5 𝑥

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