Question

Can you help me find the derivatives of these questions please

Answer 1

Answer:

(Number 5)

`(dy)/(dx)=\text{ }x(2^x)\ln 2\text{ }+2^x`Explanations:

The given equation is:

`y=x(2^x)`

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:

`(dy)/(dx)\text{ = U}(dV)/(dx)+V(dU)/(dx)`

In the equation y = x (2^x):

`\begin{gathered} U\text{ = x} \\ (dU)/(dx)=\text{ 1} \\ V=2^x \\ (dV)/(dx)=2^x\ln 2 \end{gathered}`

Substituting the U, V, dU/dx, and dV/dx into the given formula for dy/dx:

`\begin{gathered} (dy)/(dx)=x(2^x)\ln 2+2^x(1) \\ (dy)/(dx)=\text{ }x(2^x)\ln 2\text{ }+2^x \end{gathered}`