To determine

To find: The derivative of the function y=ex+1+1.

Answer

The derivative of the function y=ex+1+1 is ex+1_.

Explanation

Given:

The function, y=ex+1+1.

Formula used:

Derivative of a Constant Function:

If c is a constant function, then ddx(c)=0 (1)

The Constant Multiple Rule:

If c is a constant and f(x) is a differentiable function, then the constant multiple rule is,

ddx[c⋅f(x)]=c⋅ddxf(x) (2)

Derivative of the Natural Exponential Function:

ddx(ex)=ex (3)

The Sum Rule:

If f(x) and g(x) are both differentiable functions, then the difference rule is,

ddx[f(x)+g(x)]=ddx(f(x))+ddx(g(x)) (4)

Calculation:

The derivative of y=ex+1+1 is dydx as follows:

dydx=ddx(y) =ddx(ex+1+1)=ddx((ex×e)+1)

Apply the sum rule as shown in equation (4).

dydx=ddx(ex×e)+ddx(1)

Apply the constant multiple rule as shown in equation (2).

dydx=eddx(ex)+ddx(1)=eddx(ex)+0              [Use the derivative of a constant function]

Use the derivative of natural exponential function as shown in equation (3).

dydx=e(ex)=eex=ex+1

Therefore, the derivative of the function y=ex+1+1 is ex+1_.