Question

Multiple Choice Let \(y=u v\) be the product of the functions \(u\) and \(v .\) Find \(y^{\prime}(1)\) if \(u(1)=2, u^{\prime}(1)=3, v(1)=-1,\) and \(v^{\prime}(1)=1\) \((\mathbf{A})-4 \quad(\mathbf{B})-1 \quad(\mathbf{C}) 1, \quad(\mathbf{D}) 4 \quad(\mathbf{E})7\)

Short answer

The solution to the problem is \(-1\), therefore the correct answer choice is B.

Step-by-step solution

Apply the Product Rule

The product rule states that the derivative of the product of two functions u and v is given by \( (uv)' = u'v + uv' .\) Write down the product rule.

Substitute Known Values

Substitute the given values into the product rule formula. We are given that \(u(1)=2, u'(1)=3, v(1)=-1,\) and \(v'(1)=1\). Substituting gives: \(y'(1) = u'(1)v(1) + u(1)v'(1) = 3*(-1) + 2*1 = -3 + 2 = -1\)

Review the Answer Choices

Now that we have calculated y'(1), we can match this with one of the answer choices. The answer is -1, which corresponds to answer choice B.