Compute
𝑑
𝑑
𝑥
(
∫
5
𝑥
𝑥
2
cos
⁡
(
𝑡
2
)
𝑑
𝑡
)
.

Question

Compute 
𝑑
𝑑
𝑥
(
∫
5
𝑥
𝑥
2
cos
⁡
(
𝑡
2
)
𝑑
𝑡
)
.

BlackTom AI · Accepted Answer

We start by restating what is being asked and listing the answer choices so we can compare them carefully.
Question and options:
- Compute d/dx of the given integral expression.
- Answer options:
  1) cos(x^4) − cos(25 x^2)
  2) −2 x sin(x^4) + 5 sin(25 x^2)
  3) −2 x sin(x^2) + 2 x sin(5 x)
  4) 10 x cos(x^4)
  5) 2 x cos(x^4) − 5 cos(25 x^2)

Next, I’ll analyze each option in light of how derivatives of integrals with variable limits or integrands depending on x behave. A common tool here is the Leibniz rule: when you differentiate an integral whose upper limit is a function of x, you evaluate the integrand at the upper limit times the derivative of that limit; if the integrand itself depends on x, you may also have a term from the partial derivative of the integrand with respect to x.

Option 1: cos(x^4) − cos(25 x^2)
- This form resembles a difference of cosine terms evaluated at functions of x. If the origin were a simple composition derivative of cos(u) with u = x^4 or u = 25x^2, you would expect a factor from chain rule, which would yield terms like −sin(x^4)·4x^3 or −sin(25x^2)·50x. Instead, this option shows cos terms, not sin, and lacks any chain-rule-derived x-factors. Moreover, if the derivative came from evaluating an integrand at an upper limit, you’d typically see sin or cos with x in a linear or cubic fashion multiplied by a derivative of the limit. The mismatch in both the functional form (cos rather than sin) and the missing x-dependence in the coefficients makes this option unlikely as the correct derivative.

Option 2: −2 x sin(x^4) + 5 sin(25 x^2)
- This expression has two distinct parts: a term involving sin(x^4) multiplied by −2x, and another term 5 sin(25 x^2). The derivative structure here is reminiscent of a combination of contributions from a product of x with a function of x, and from an integral with an x-dependent upper limit where the integrand evaluated at that limit yields a sine term. Specifically, the −2x factor multiplying sin(x^4) matches a chain-rule type derivative where the inner argument is x^4 and the derivative of x^4 is 4x, combined with another balancing factor that yields −2x. The 5 sin(25 x^2) piece mirrors the possibility of an upper-limit contribution or a partial-derivative with respect to x inside the integrand that yields sin(25x^2). Overall, the combination fits a common pattern for derivatives of integrals with x appearing inside both the limit and the integrand.

Option 3: −2 x sin(x^2) + 2 x sin(5 x)
- This option presents terms with sin(x^2) and sin(5x) and linear x factors. While a derivative involving sin of an inner function is plausible, the specific inner arguments here (x^2 and 5x) do not align naturally with the typical inner structure that would arise from an integrand like cos(t^2) or sin(t^2) integrated with respect to t and differentiated with respect to x. Moreover, the combination 2x sin(5x) suggests a derivative coming from a limit involving 5x, but pairing that with −2x sin(x^2) without a consistent linking mechanism (such as a single variable change or a coherent upper limit) makes this option less coherent with a standard Leibniz-rule-based derivative of a single-variable integral.

Option 4: 10 x cos(x^4)
- This term looks like a chain-rule result of a derivative with an inner function x^4 inside a cosine, scaled by the outer derivative. However, if we were differentiating something like ∫_0^{x^2} f(t) dt and f(t) involved a cos(t^2) or similar, you would typically see a factor with cos(x^4) multiplied by the derivative of the upper limit (which could be 2x or 4x, depending on the exact upper limit). Seeing 10x cos(x^4) specifically would require a very particular setup (like an upper limit that yields 5 times the derivative 2x, or an integrand contributing an extra factor of 5). Without that precise construction, this option seems mismatched to the standard derivative patterns for the given integral form.

Option 5: 2 x cos(x^4) − 5 cos(25 x^2)
- Here we have a cos(x^4) term with a 2x factor and another cosine term with 25x^2 multiplied by −5. This resembles a combination of a chain-rule contribution for cos(x^4) and another upper-limit or integrand-derived cosine term. Yet the signs and coefficients do not align with the sine-based outcomes that are typical when differentiating integrals of the type ∫ f(t) dt with t appearing inside a cosine of t^2 or similar. The presence of cos rather than sin in the second term and the specific coefficients make this option seem inconsistent with the standard derivative patterns for the problem as usually presented.

Putting the pieces together, Option 2 stands out as the one whose structure best matches the expected combination of a term arising from differentiating an x-dependent inner part (giving the −2x sin(x^4) piece) and a term arising from the contribution of the integrand evaluated at an x-dependent limit (giving the 5 sin(25 x^2) piece). The other options either mismatch in trigonometric function (cos vs sin), have incompatible inner arguments, or present coefficient patterns that do not align with the typical Leibniz-rule-derived derivatives for this kind of integral.

Additional note on approach and pitfalls:
- Remember that derivatives of integrals with variable limits often produce terms like f(b(x))·b'(x). If the integrand also depends on x, you may have an extra ∂/∂x f(x, t) term inside the integral. In many standard exercises of this type, the final result is a sum of terms where one term comes from evaluating the integrand at the moving upper limit, and another comes from differentiating an x-dependent part of the integrand itself.
- Misconceptions to watch out for include treating all terms as coming from a single application of the fundamental theorem of calculus without accounting for possible x-dependence in the integrand, or mixing up sin and cos without tracking chain-rule factors that arise from inner functions like x^4 or 25x^2.

Thus, the reasoning above supports option 2 as the one most consistent with the derivative form expected from the given integral, with the other options presenting inconsistencies in trigonometric form, inner functions, or coefficient structure.

MAT137Y1 LEC 20249: Calculus with Proofs (all lecture sections) Pre-Class Quiz 42(8.3 and 8.4)

Compute 𝑑 𝑑 𝑥 ( ∫ 5 𝑥 𝑥 2 cos ⁡ ( 𝑡 2 ) 𝑑 𝑡 ) .

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