Answer :
Answer:
(Number 5)
[tex]\frac{dy}{dx}=\text{ }x(2^x)\ln 2\text{ }+2^x[/tex]Explanations:The given equation is:
[tex]y=x(2^x)[/tex]This function represents the product of x and 2^x, therefore, to find the derivative, we will use the product rule.
When y = UV, dy/dx is given as:
[tex]\frac{dy}{dx}\text{ = U}\frac{dV}{dx}+V\frac{dU}{dx}[/tex]In the equation y = x (2^x):
[tex]\begin{gathered} U\text{ = x} \\ \frac{dU}{dx}=\text{ 1} \\ V=2^x \\ \frac{dV}{dx}=2^x\ln 2 \end{gathered}[/tex]Substituting the U, V, dU/dx, and dV/dx into the given formula for dy/dx:
[tex]\begin{gathered} \frac{dy}{dx}=x(2^x)\ln 2+2^x(1) \\ \frac{dy}{dx}=\text{ }x(2^x)\ln 2\text{ }+2^x \end{gathered}[/tex]