Solve (-x^4+10x^3+5x)+(-6x^4-8x^3+2)

$$( - x ^ { 4 } + 10 x ^ { 3 } + 5 x ) + ( - 6 x ^ { 4 } - 8 x ^ { 3 } + 2 )$$

Evaluate

$$2+5x+2x^{3}-7x^{4}$$

Solution Steps

Combine 10x^{3} and -8x^{3} to get 2x^{3}.

$$-x^{4}+2x^{3}+5x-6x^{4}+2$$

Combine -x^{4} and -6x^{4} to get -7x^{4}.

$$-7x^{4}+2x^{3}+5x+2$$

Differentiate w.r.t. x

$$5+6x^{2}-28x^{3}$$

Steps Using Derivative Rule for Sum

Combine 10x^{3} and -8x^{3} to get 2x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-x^{4}+2x^{3}+5x-6x^{4}+2)$$

Combine -x^{4} and -6x^{4} to get -7x^{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-7x^{4}+2x^{3}+5x+2)$$

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}.

$$4\left(-7\right)x^{4-1}+3\times 2x^{3-1}+5x^{1-1}$$

Multiply 4 times -7.

$$-28x^{4-1}+3\times 2x^{3-1}+5x^{1-1}$$

Subtract 1 from 4.

$$-28x^{3}+3\times 2x^{3-1}+5x^{1-1}$$

Multiply 3 times 2.

$$-28x^{3}+6x^{3-1}+5x^{1-1}$$

Subtract 1 from 3.

$$-28x^{3}+6x^{2}+5x^{1-1}$$

Subtract 1 from 1.

$$-28x^{3}+6x^{2}+5x^{0}$$

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

$$-28x^{3}+6x^{2}+5\times 1$$

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

$$-28x^{3}+6x^{2}+5$$