Find the derivative.

Question

BlackTom AI · Accepted Answer

We start by identifying the given expression: the derivative with respect to x of the integral from 1 to sqrt(x) of 14 t^9 dt.
Option A: 7/5 x^6 − 7/5. This is not correct because after evaluating the integral with upper limit u = sqrt(x), we get a function involving u^10, which becomes x^5 when substituted, not something like x^6. The derivative would then involve x^4, not x^6.
Option B: 14x^9/2. This seems to come from applying the power rule incorrectly inside the derivative, but the correct substitution u = sqrt(x) yields x^5 terms after integration, so the derivative should not be proportional to x^9 or involve a factor 14/2 in this context.
Option C: 7x^4. This matches the correct differentiation result. Indeed, compute the integral: ∫_1^{√x} 14 t^9 dt = 14 [t^10/10]_1^{√x} = (7/5)[(√x)^{10} − 1] = (7/5)(x^5 − 1). Differentiate with respect to x: d/dx[(7/5)(x^5 − 1)] = (7/5) · 5x^4 = 7x^4. Therefore this option is correct.
Option D: 28/3 x^6. This is not correct because it implies a derivative proportional to x^6, which would result from a different antiderivative form than the one obtained from substituting u = √x into the evaluated integral. The actual derivative yields x^4 terms, not x^6.
In summary, only Option C yields 7x^4 after performing the substitution u = √x, evaluating the definite integral, and differentiating with respect to x. The other choices arise from misapplying the limits, wrong exponent manipulations, or incorrect differentiation of the resulting expression.

MATH2415W16 Quiz 4 Dec 16 (W16)- Requires Respondus LockDown Browser

Find the derivative.

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MATH*2415*W16 Quiz 4 Dec 16 (W16)- Requires Respondus LockDown Browser

Find the derivative.

View Explanation

Log in for full answers

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Let [math: F(x)=∫03xet2 dt]\displaystyle F(x)=\int _0^{3x}{e^{t^2} dt}. Compute [math: F′(0.5)]F’(0.5). (Correct the answer to 2 decimal places.)

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MATH2415W16 Quiz 4 Dec 16 (W16)- Requires Respondus LockDown Browser