Find the derivative using Formula (18), and check your...

Find the derivative using Formula (18), and check your answer by evaluating the integral and then differentiating the result.
$$
\text { (a) } \frac{d}{d x} \int_{1}^{x^{3}} \frac{1}{t} d t \quad \text { (b) } \frac{d}{d x} \int_{1}^{\ln x} e^{t} d t
$$

Key Concepts

Verification by Differentiation

This technique involves computing an antiderivative of an integrand to evaluate a definite integral and then differentiating the result to confirm that it matches the directly computed derivative. Using this process provides a check for the correctness of the derivative computed via formulae such as the one derived from the Fundamental Theorem of Calculus.

Fundamental Theorem of Calculus

This theorem establishes the connection between integration and differentiation. It states that if a function is continuous over an interval, then the integral of that function from a fixed lower limit to a variable upper limit is differentiable, and its derivative equals the original function evaluated at the upper limit.

Chain Rule

The chain rule is a technique used for differentiating composite functions. In problems involving integrals with variable limits, the chain rule is applied to account for the fact that the upper limit is itself a function of the variable of differentiation, requiring multiplication by the derivative of that limit.

Differentiation of Integrals with Variable Limits

This concept extends the Fundamental Theorem of Calculus to integrals whose limits are functions of the variable. It involves taking the derivative of an integral by evaluating the integrand at the variable limit and then multiplying by the derivative of that limit, thereby combining ideas from both the Fundamental Theorem of Calculus and the chain rule.

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