Solve (9h^3+h^2+2h+3)+(5h^2+2h+9)

$$( 9 h ^ { 3 } + h ^ { 2 } + 2 h + 3 ) + ( 5 h ^ { 2 } + 2 h + 9 )$$

Evaluate

$$9h^{3}+6h^{2}+4h+12$$

Solution Steps

Combine h^{2} and 5h^{2} to get 6h^{2}.

$$9h^{3}+6h^{2}+2h+3+2h+9$$

Combine 2h and 2h to get 4h.

$$9h^{3}+6h^{2}+4h+3+9$$

Add 3 and 9 to get 12.

$$9h^{3}+6h^{2}+4h+12$$

Differentiate w.r.t. h

$$27h^{2}+12h+4$$

Steps Using Derivative Rule for Sum

Combine h^{2} and 5h^{2} to get 6h^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}h}(9h^{3}+6h^{2}+2h+3+2h+9)$$

Combine 2h and 2h to get 4h.

$$\frac{\mathrm{d}}{\mathrm{d}h}(9h^{3}+6h^{2}+4h+3+9)$$

Add 3 and 9 to get 12.

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

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}.

$$3\times 9h^{3-1}+2\times 6h^{2-1}+4h^{1-1}$$

Multiply 3 times 9.

$$27h^{3-1}+2\times 6h^{2-1}+4h^{1-1}$$

Subtract 1 from 3.

$$27h^{2}+2\times 6h^{2-1}+4h^{1-1}$$

Multiply 2 times 6.

$$27h^{2}+12h^{2-1}+4h^{1-1}$$

Subtract 1 from 2.

$$27h^{2}+12h^{1}+4h^{1-1}$$

Subtract 1 from 1.

$$27h^{2}+12h^{1}+4h^{0}$$

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

$$27h^{2}+12h+4h^{0}$$

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

$$27h^{2}+12h+4\times 1$$

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

$$27h^{2}+12h+4$$