Question

derivative of integrally defined functions
this is the only question in this section.
question
evaluate \\( \frac{d}{dx} \int_{1}^{\sqrt{x}} \sin(5t) \\, dt \\)

Explanation:

Step1: Apply Leibniz Rule

The Leibniz rule for differentiation under the integral sign states that if \( F(x)=\int_{a(x)}^{b(x)} f(t) dt \), then \( F^\prime(x)=f(b(x))\cdot b^\prime(x)-f(a(x))\cdot a^\prime(x) \). Here, \( a(x) = 1 \) (so \( a^\prime(x)=0 \)), \( b(x)=\sqrt{x} \), and \( f(t)=\sin(5t) \).

Step2: Find \( b^\prime(x) \)

First, find the derivative of \( b(x)=\sqrt{x}=x^{\frac{1}{2}} \). Using the power rule, \( b^\prime(x)=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}} \).

Step3: Substitute into Leibniz rule

Since \( a^\prime(x) = 0 \), we have \( F^\prime(x)=f(b(x))\cdot b^\prime(x)-f(a(x))\cdot0=f(\sqrt{x})\cdot\frac{1}{2\sqrt{x}} \). Substitute \( t = \sqrt{x} \) into \( f(t)=\sin(5t) \), so \( f(\sqrt{x})=\sin(5\sqrt{x}) \). Then \( F^\prime(x)=\sin(5\sqrt{x})\cdot\frac{1}{2\sqrt{x}} \).

Answer:

\(\frac{\sin(5\sqrt{x})}{2\sqrt{x}}\)