Question

find the derivative.

$$\frac{d}{dx}\int_{0}^{\sqrt{x}}\cos 11t dt$$

a. by evaluating the integral and differentiating the result.
b. by differentiating the integral directly.

then use the product rule of differentiation.

use the fundamental theorem of calculus, part 1, with 0 as lower limit and $$\sqrt{x}$$ as up
then use the chain rule of differentiation.

differentiate the integral directly.

$$\frac{d}{dx}\int_{0}^{\sqrt{x}}\cos 11t dt=\square$$

(simplify your answer. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Apply the Fundamental Theorem of Calculus Part 1 and Chain Rule

Let \(u = \sqrt{x}\), then \(\frac{d}{dx}\int_{0}^{\sqrt{x}}\cos(11t)dt\).
By the Fundamental Theorem of Calculus Part 1: If \(F(x)=\int_{a}^{x}f(t)dt\), then \(F^\prime(x)=f(x)\). Here, if \(y = \int_{0}^{u}\cos(11t)dt\), then \(\frac{dy}{du}=\cos(11u)\).
And by the chain rule \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\).
Since \(u = \sqrt{x}=x^{\frac{1}{2}}\), then \(\frac{du}{dx}=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}\).

Step2: Substitute \(u = \sqrt{x}\) into the expression

Substitute \(u=\sqrt{x}\) into \(\frac{dy}{du}\cdot\frac{du}{dx}\).
We get \(\frac{dy}{dx}=\cos(11\sqrt{x})\cdot\frac{1}{2\sqrt{x}}\).

Answer:

\(\frac{\cos(11\sqrt{x})}{2\sqrt{x}}\)