Exercise · ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-015 · ★★☆ Intermediate
Compute \(\displaystyle\frac{d}{dx}\int_0^{x^2}e^{t}\,dt\).
Choose one option.
Show answer
- B. \(2x\,e^{x^2}\)
Explanation
Chain rule: the derivative is \(e^{x^2}\cdot\tfrac{d}{dx}(x^2)=2x\,e^{x^2}\). The answer \(e^{x^2}\) forgets the inner factor.
Review the concept: Fundamental theorem of calculus
It links derivatives and integrals: a definite integral equals an antiderivative evaluated at the endpoints, and differentiating an integral with variable upper limit returns the integrand.
\[\begin{gathered}\int_a^b f(x)\,dx=F(b)-F(a)\\ \frac{d}{dx}\int_a^x f(t)\,dt=f(x)\end{gathered}\]
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