#### To determine

**To find:** The derivative of f(x)=5x+1.

#### Answer

The derivative of f(x)=5x+1 is 525x+1.

#### Explanation

**Given:**

The function is f(x)=5x+1.

**Derivative rule: The Power Rule combined with the Chain Rule**

If *n* is any real number and g(x) is differentiable function, then

ddx[g(x)]n=n[g(x)]n−1g′(x) (1)

**Calculation:**

Obtain the derivative of f(x)*.*

f′(x)=ddx(f(x))=ddx(5x+1)=ddx((5x+1)12)

Here, g(x)=5x+1.

Apply the Power Rule combined with the chain rule as shown in equation (1),

f′(x)=ddx(5x+1)12=12(5x+1)12−1⋅ddx(5x+1)=12(5x+1)−12(5ddx(x)+ddx(1))=12(5x+1)−12(5(1x1−1)+(0))

Simplify further and obtain the derivative of the function.

f′(x)=12(5x+1)−12(5(1)) =52(5x+1)−12=52(5x+1)12=52(5x+1)

Therefore, the derivative of f(x)=5x+1 is 525x+1_.