Mathematics

Mathematics

Problem

Let F(x)=∫2x3+1(4t2−1) dtF(x)=\displaystyle\int_{2}^{x^{3}+1}\bigl(4t^{2}-1\bigr)\,dt. What is F′(x)F'(x) ? a) 4(x3+1)2−14(x^{3}+1)^{2}-1 b) (4x2−1) 3x2\bigl(4x^{2}-1\bigr)\,3x^{2} c) (4(x3+1)2−1) 3x2\bigl(4(x^{3}+1)^{2}-1\bigr)\,3x^{2} d) (4(x3+1)2−1) (x3+1)\bigl(4(x^{3}+1)^{2}-1\bigr)\,(x^{3}+1)
Intuitive explanation

The key is recognizing this as a composition: you have an integral with a variable upper limit that itself depends on xx. Use the Fundamental Theorem of Calculus combined with the chain rule—differentiate the outer integral with respect to its upper limit, then multiply by the derivative of that limit with respect to xx. This two-step process captures how changes in xx propagate through the nested functions.

Intuitive explanation

The key is recognizing this as a composition: you have an integral with a variable upper limit that itself depends on xx. Use the Fundamental Theorem of Calculus combined with the chain rule—differentiate the outer integral with respect to its upper limit, then multiply by the derivative of that limit with respect to xx. This two-step process captures how changes in xx propagate through the nested functions.

Browse the problem library