‘Calculate the derivative 10x In (t)dt dx 29 using Part 2 of the Fundamental Theorem of Calculus_ Enclose arguments of functions in parentheses_ For example; sin (2x) 10x In (t)dt = dx 29’
The Correct Answer and Explanation is:
We are asked to calculate the derivative of the integral∫10x29ln(t) dtwith respect to x.\int_{10x}^{29} \ln(t) \, dt \quad \text{with respect to} \, x.∫10×29ln(t)dtwith respect tox.
This is a problem that can be approached using Part 2 of the Fundamental Theorem of Calculus. Part 2 of this theorem states that if F(x)=∫a(x)b(x)f(t) dtF(x) = \int_{a(x)}^{b(x)} f(t) \, dtF(x)=∫a(x)b(x)f(t)dt, then the derivative of F(x)F(x)F(x) with respect to xxx is given by:ddxF(x)=f(b(x))⋅ddxb(x)−f(a(x))⋅ddxa(x).\frac{d}{dx} F(x) = f(b(x)) \cdot \frac{d}{dx} b(x) – f(a(x)) \cdot \frac{d}{dx} a(x).dxdF(x)=f(b(x))⋅dxdb(x)−f(a(x))⋅dxda(x).
Step-by-Step Solution
- Identify the functions involved:
- The integrand is ln(t)\ln(t)ln(t).
- The limits of integration are a(x)=10xa(x) = 10xa(x)=10x and b(x)=29b(x) = 29b(x)=29.
- Apply the formula:
- The derivative of the upper limit b(x)=29b(x) = 29b(x)=29 with respect to xxx is 0 (since it is a constant).
- The derivative of the lower limit a(x)=10xa(x) = 10xa(x)=10x with respect to xxx is 10.
Thus, by Part 2 of the Fundamental Theorem of Calculus, we get:ddx(∫10x29ln(t) dt)=ln(29)⋅0−ln(10x)⋅10.\frac{d}{dx} \left( \int_{10x}^{29} \ln(t) \, dt \right) = \ln(29) \cdot 0 – \ln(10x) \cdot 10.dxd(∫10×29ln(t)dt)=ln(29)⋅0−ln(10x)⋅10.
Since the term involving ln(29)\ln(29)ln(29) vanishes (because of the zero derivative of the upper limit), the result simplifies to:−10ln(10x).-10 \ln(10x).−10ln(10x).
Conclusion
The derivative of the integral is:ddx(∫10x29ln(t) dt)=−10ln(10x).\frac{d}{dx} \left( \int_{10x}^{29} \ln(t) \, dt \right) = -10 \ln(10x).dxd(∫10×29ln(t)dt)=−10ln(10x).
This result follows directly from the application of Part 2 of the Fundamental Theorem of Calculus. We simply evaluated the integrand at the limits of integration and multiplied by the derivative of the respective limits.
