Question

Use the given information to find \(f^{\prime}(2)\). \(g(2)=3 \quad\) and \(\quad g^{\prime}(2)=-2\) \(h(2)=-1 \quad\) and \(\quad h^{\prime}(2)=4\) $$ f(x)=2 g(x)+h(x) $$

Short answer

The derivative of \(f\) at \(x = 2\) is \(f^{\prime}(2) = 0\).

Step-by-step solution

Differentiate Function \(f(x)\)

Firstly, we need to differentiate \(f(x)\) with respect to \(x\). Given that \(f(x) = 2g(x) + h(x)\), by applying the basic rules of differentiation to each term, we find that \(f^{\prime}(x) = 2g^{\prime}(x) + h^{\prime}(x)\). This rule states that the derivative of a sum of functions is the sum of their derivatives.

Apply Given Values

Now that we have differentiated the function \(f(x)\), we can evaluate it at \(x = 2\). That is, we substitute \(x = 2\) into the derivative, using \(g^{\prime}(2) = -2\) and \(h^{\prime}(2) = 4\). So, \(f^{\prime}(2) = 2(-2) + 4 = -4 + 4\)

Simplify Expression

Finally, we simplify the expression obtained in the previous step, we find that \(f^{\prime}(2) = 0\)