To determine

To find: The derivative of the function g(x).

Answer

The derivative of the function g(x)=74x2−3x+12.is 72x−3.

Explanation

Given:

The function, g(x)=74x2−3x+12.

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)

The Power Rule:

If n is any real number, then the power rule is,

ddx(xn)=nxn−1 (3)

The Sum Rule:

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

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

The Difference Rule:

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

          ddx[f(x)−g(x)]=ddx(f(x))−ddx(g(x)) (5)

Calculation:

The derivative of g(x)=74x2−3x+12 is g′(x) as follows,

g′(x)=ddx(g(x)) =ddx(74x2−3x+12)

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

g′(x)   =ddx(74x2−3x)+ddx(12) 

Apply the difference rule as shown equation (5).

g′(x) =ddx(74x2)−ddx(3x)+ddx(12) 

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

g′(x)=74ddx(x2)−3ddx(x)+ddx(12) 

Apply the power rule as shown in equation (3).

g′(x)=74(2x2−1)−3(1x1−1)+ddx(12) 

In equation (1), substitute 12 for c.

g′(x)=74(2x1)−3(1x0)+0          =74×2×x−3(x0)+0=72x−3(1)=72x−3

Therefore, the derivative of the function g(x)=74x2−3x+12.is 72x−3.