Question

Use part 1 of the fundamental theorem of calculus to find the derivative of the function. y = 5 u3 1 + u2 du 4 − 3x

Answer 1

Looks like

$y(x)=\displaystyle\int_5^{4-3x}u^3(1+u^2)\,\mathrm du$

in which case the FTC asserts that

$\dfrac{\mathrm dy}{\mathrm dx}=(4-3x)^3(1+(4-3x)^2)\cdot\dfrac{\mathrm d(4-3x)}{\mathrm dx}$

$\dfrac{\mathrm dy}{\mathrm dx}=-3(4-3x)^3(1+(4-3x)^2)$

Answer 2

Using part 1 of the fundamental theorem of calculus to find the derivative of the function. The derivative of the given function is:

$\mathbf{\dfrac{dy}{dx} =3\Big (\dfrac{(4-3x)^3}{1+(4-3x)^2}\ \Big) \ }$

Consider the given function:

$\mathbf{y = \int^5_{4-3x} \ \dfrac{u^3}{1+u^2}\ du}$

The objective is to find  $\mathbf{\dfrac{dy}{dx}}$  by using the fundamental theorem of calculus.

• Suppose v = 4 - 3x;
• Then dv = -3dx
• $\mathbf{\dfrac{dv}{dx}= -3}$  

Using chain rule:

$\mathbf{\dfrac{dy}{dx} = \dfrac{dy}{dv}\times \dfrac{dv}{dx}}$

$\mathbf{ =\dfrac{d}{dv}\Big (\int^5_{4-3x} \dfrac{u^3}{1+u^2}\ du\Big) \dfrac{dv}{dx}}}$

$\mathbf{ =\dfrac{d}{dv}\Big (\int^5_{v} \dfrac{u^3}{1+u^2}\ du\Big) \dfrac{dv}{dx} \ \ \ \ \ since \ v \ = 4 - 3x} }$

$\mathbf{ =-\dfrac{d}{dv}\Big (\int^v_{5} \dfrac{u^3}{1+u^2}\ du\Big) \dfrac{dv}{dx} }$

$\mathbf{ =-\dfrac{d}{dv}\Big (\int^v_{5} \dfrac{u^3}{1+u^2}\ du\Big) \ (-3)}$

$\mathbf{ =3\dfrac{d}{dv}\Big (\int^v_{5} \dfrac{u^3}{1+u^2}\ du\Big) \ }$

From the fundamental theorem of calculus;

 $\mathbf{\dfrac{d}{dx} \Big( \int^x_1 \ g(t) dt \Big) = g(x)}$

∴

$\mathbf{ =3\dfrac{d}{dv}\Big (\int^v_{5} \dfrac{u^3}{1+u^2}\ du\Big) \ }$ will be:

$\mathbf{ =3\times \Big (\dfrac{(4-3x)^3}{1+(4-3x)^2}\ \Big) \ }$

∴

$\mathbf{\dfrac{dy}{dx} =3\Big (\dfrac{(4-3x)^3}{1+(4-3x)^2}\ \Big) \ }$

Therefore, we can conclude that the derivative of $\mathbf{y = \int^5_{4-3x} \ \dfrac{u^3}{1+u^2}\ du}$

using the fundamental theorem of calculus is   $\mathbf{\dfrac{dy}{dx} =3\Big (\dfrac{(4-3x)^3}{1+(4-3x)^2}\ \Big) \ }$

Learn more about fundamental derivatives here:

https://brainly.com/question/9964510?referrer=searchResults