Solve (6x^4-3x^3+9x)+(8x^4+10x^3-4)

$$( 6 x ^ { 4 } - 3 x ^ { 3 } + 9 x ) + ( 8 x ^ { 4 } + 10 x ^ { 3 } - 4 )$$

Evaluate

$$14x^{4}+7x^{3}+9x-4$$

Solution Steps

Combine 6x^{4} and 8x^{4} to get 14x^{4}.

$$14x^{4}-3x^{3}+9x+10x^{3}-4$$

Combine -3x^{3} and 10x^{3} to get 7x^{3}.

$$14x^{4}+7x^{3}+9x-4$$

Differentiate w.r.t. x

$$56x^{3}+21x^{2}+9$$

Steps Using Derivative Rule for Sum

Combine 6x^{4} and 8x^{4} to get 14x^{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(14x^{4}-3x^{3}+9x+10x^{3}-4)$$

Combine -3x^{3} and 10x^{3} to get 7x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(14x^{4}+7x^{3}+9x-4)$$

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}.

$$4\times 14x^{4-1}+3\times 7x^{3-1}+9x^{1-1}$$

Multiply 4 times 14.

$$56x^{4-1}+3\times 7x^{3-1}+9x^{1-1}$$

Subtract 1 from 4.

$$56x^{3}+3\times 7x^{3-1}+9x^{1-1}$$

Multiply 3 times 7.

$$56x^{3}+21x^{3-1}+9x^{1-1}$$

Subtract 1 from 3.

$$56x^{3}+21x^{2}+9x^{1-1}$$

Subtract 1 from 1.

$$56x^{3}+21x^{2}+9x^{0}$$

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

$$56x^{3}+21x^{2}+9\times 1$$

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

$$56x^{3}+21x^{2}+9$$