Question

Suppose \(u\) and \(v\) are functions of \(x\) that are differentiable at \(x=2\) and that \(u(2)=3, u^{\prime}(2)=-4, v(2)=1,\) and \(v^{\prime}(2)=2 .\) Find the values of the following derivatives at \(x=2\) (a) \(\frac{d}{d x}(u v)\) (b) \(\frac{d}{d x}\left(\frac{u}{v}\right)\) (c) \(\frac{d}{d x}\left(\frac{v}{u}\right)\) (d) \(\frac{d}{d x}(3 u-2 v+2 u v)\)

Short answer

The values of the derivatives at \(x=2\) are: (a) 2 , (b) -10 , (c) \(10/9\) , and (d) -12.

Step-by-step solution

Find derivative of \(u*v\)

The derivative of the product of two functions \(u\) and \(v\) can be found by applying the product rule. Therefore, \(\frac{d}{d x}(u v) = u^{\prime}(x) * v(x) + u(x) * v^{\prime}(x)\). Substituting the given values at \(x=2\), the derivative becomes \(-4*1 + 3*2 = -4 + 6 = 2\).

Find derivative of \(\frac{u}{v}\)

The derivative of the quotient of two functions \(u\) and \(v\) can be found using the quotient rule. Therefore, \(\frac{d}{d x}\left(\frac{u}{v}\right) = \frac{v(x) * u^{\prime}(x) - u(x) * v^{\prime}(x)}{[v(x)]^2}\). Substituting the given values at \(x=2\), the derivative becomes \(\frac{1*(-4) - 3*2}{(1)^2} = -4 - 6 = -10\).

Find derivative of \(\frac{v}{u}\)

Now, we'll find derivative of the quotient of the functions \(v\) and \(u\), using the quotient rule. So, \(\frac{d}{d x}\left(\frac{v}{u}\right) = \frac{u(x) * v^{\prime}(x) - v(x) * u^{\prime}(x)}{[u(x)]^2}\). With the given values at \(x=2\), the derivative becomes \(\frac{3*2 - 1*(-4)}{(3)^2} = \frac{10}{9}\).

Find derivative of \(3u-2v+2uv\)

The derivative of a sum of functions is the sum of their derivatives. So, we distribute the derivative across the terms. The derivative of \(3 u\) is \(3 u^{\prime}(x)\), of \(-2 v\) is \(-2 v^{\prime}(x)\), and for \(2uv\) apply the product rule. Therefore, \(\frac{d}{d x}(3u-2v+2uv) = 3 u^{\prime}(x) - 2 v^{\prime}(x) + 2[u^{\prime}(x) * v(x) + u(x) * v^{\prime}(x)]\). Substituting the given values at \(x=2\), the derivative becomes \(3*(-4) - 2*2 + 2[(-4)*1 + 3*2] = -12 - 4 + 2[-4 + 6] = -16 + 4 = -12\).