Question

Suppose \(u\) and \(v\) are functions of \(x\) that are differentiable at \(x=0,\) and that \(u(0)=5, u^{\prime}(0)=-3, v(0)=-1, v^{\prime}(0)=2\) Find the values of the following derivatives at \(x=0\) (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) \(\frac{d}{d x}(7 v-2 u)\)

Short answer

(a) The derivative of \(u v\) at \(x=0\) is 13. (b) The derivative of \(\frac{u}{v}\) at \(x=0\) is 13. (c) The derivative of \(\frac{v}{u}\) at point \(x=0\) is \(\frac{7}{25}\). (d) The derivative of \(7v-2u\) is 20 at \(x=0\).

Step-by-step solution

Calculate the derivative of \(u v\)

Apply the product rule for derivatives \(\frac{d(uv)}{dx} = u^{\prime}v+uv^{\prime}\). At point \(x=0\), this becomes \(-3*-1+5*2 = 3+10 = 13\)

Calculate the derivative of \(\frac{u}{v}\)

Apply the quotient rule \(\frac{d(u/v)}{dx} = \frac{v*u^{\prime}-u*v^{\prime}}{v^2}\). At point \(x=0\), this becomes \(\frac{-1*-3-5*2}{(-1)^2} = \frac{13}{1} = 13\)

Calculate the derivative of \(\frac{v}{u}\)

Apply the quotient rule again \(\frac{d(v/u)}{dx} = \frac{u*v^{\prime}-v*u^{\prime}}{u^2}\). At point \(x=0\), this becomes \(\frac{5*2-(-1*-3)}{5^2} = \frac{7}{25}\)

Calculate the derivative of \(7v-2u\)

To find the derivative of this expression, apply the derivative rule for addition/subtraction of functions and the constant multiple rule to obtain \(7v^{\prime}-2u^{\prime}\). At point \(x=0\), this becomes \(7*2-2*-3 = 14+6 = 20\)