Worked problem · Calculus I · Units 1–3A

Fundamental Theorem Derivative: Worked Example and Solution

A complete course synthesis example with method choice, derivation, verification, and a common wrong approach.

1 problem8 min estimated timeCumulative progression

What is included

See a concise answer and full derivation for differentiate $f(x)=\int_1^x\cos(t^2)dt$.

Skills assessed

Course synthesis

Prerequisites

Calculus I Units 1 through 3A

Printable preview

Differentiate $F(x)=\int_1^x\cos(t^2)dt$ .

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

Problem 1: Differentiate $F(x)=\int_1^x\cos(t^2)dt$ .

Answer: $\cos(x^2)$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Apply FTC Part I.
The verified result is $\cos(x^2)$ .

Common errors

Selecting a familiar formula without checking its hypotheses.

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

Problem 1: Differentiate $F(x)=\int_1^x\cos(t^2)dt$ .

Answer: $\cos(x^2)$

Why this method: Course synthesis matches the mathematical structure before any algebraic cleanup.

Apply FTC Part I.
The verified result is $\cos(x^2)$ .

What is included

Skills assessed

Prerequisites

Printable preview

Complete worked solutions

Problem 1: Differentiate F(x)=∫1xcos⁡(t2)dtF(x)=\int_1^x\cos(t^2)dtF(x)=∫1x​cos(t2)dt.

Common errors

Continue learning

Lessons and articles

Related resources

Glossary

What is included

Skills assessed

Prerequisites

Printable preview

Complete worked solutions

Problem 1: Differentiate F(x)=∫1xcos⁡(t2)dtF(x)=\int_1^x\cos(t^2)dtF(x)=∫1x​cos(t2)dt.

Common errors

Continue learning

Lessons and articles

Related resources

Glossary

Problem 1: Differentiate $F(x)=\int_1^x\cos(t^2)dt$ .

Problem 1: Differentiate $F(x)=\int_1^x\cos(t^2)dt$ .