Solve (2m^3+m^2+8m+9)+(9m^3-5m^2+4m+6)

$$( 2 m ^ { 3 } + m ^ { 2 } + 8 m + 9 ) + ( 9 m ^ { 3 } - 5 m ^ { 2 } + 4 m + 6 )$$

Evaluate

$$11m^{3}-4m^{2}+12m+15$$

Solution Steps

Combine 2m^{3} and 9m^{3} to get 11m^{3}.

$$11m^{3}+m^{2}+8m+9-5m^{2}+4m+6$$

Combine m^{2} and -5m^{2} to get -4m^{2}.

$$11m^{3}-4m^{2}+8m+9+4m+6$$

Combine 8m and 4m to get 12m.

$$11m^{3}-4m^{2}+12m+9+6$$

Add 9 and 6 to get 15.

$$11m^{3}-4m^{2}+12m+15$$

Differentiate w.r.t. m

$$33m^{2}-8m+12$$

Steps Using Derivative Rule for Sum

Combine 2m^{3} and 9m^{3} to get 11m^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(11m^{3}+m^{2}+8m+9-5m^{2}+4m+6)$$

Combine m^{2} and -5m^{2} to get -4m^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(11m^{3}-4m^{2}+8m+9+4m+6)$$

Combine 8m and 4m to get 12m.

$$\frac{\mathrm{d}}{\mathrm{d}m}(11m^{3}-4m^{2}+12m+9+6)$$

Add 9 and 6 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}m}(11m^{3}-4m^{2}+12m+15)$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$3\times 11m^{3-1}+2\left(-4\right)m^{2-1}+12m^{1-1}$$

Multiply 3 times 11.

$$33m^{3-1}+2\left(-4\right)m^{2-1}+12m^{1-1}$$

Subtract 1 from 3.

$$33m^{2}+2\left(-4\right)m^{2-1}+12m^{1-1}$$

Multiply 2 times -4.

$$33m^{2}-8m^{2-1}+12m^{1-1}$$

Subtract 1 from 2.

$$33m^{2}-8m^{1}+12m^{1-1}$$

Subtract 1 from 1.

$$33m^{2}-8m^{1}+12m^{0}$$

For any term t, t^{1}=t.

$$33m^{2}-8m+12m^{0}$$

For any term t except 0, t^{0}=1.

$$33m^{2}-8m+12\times 1$$

For any term t, t\times 1=t and 1t=t.

$$33m^{2}-8m+12$$