Solve (3x^2-5x+7)+(-2+6x-7x^2+x^3)

$$( 3 x ^ { 2 } - 5 x + 7 ) + ( - 2 + 6 x - 7 x ^ { 2 } + x ^ { 3 } )$$

Evaluate

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

Solution Steps

Subtract 2 from 7 to get 5.

$$3x^{2}-5x+5+6x-7x^{2}+x^{3}$$

Combine -5x and 6x to get x.

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

Combine 3x^{2} and -7x^{2} to get -4x^{2}.

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

Differentiate w.r.t. x

$$3x^{2}-8x+1$$

Steps Using Derivative Rule for Sum

Subtract 2 from 7 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{2}-5x+5+6x-7x^{2}+x^{3})$$

Combine -5x and 6x to get x.

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

Combine 3x^{2} and -7x^{2} to get -4x^{2}.

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

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}.

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

Multiply 2 times -4.

$$-8x^{2-1}+x^{1-1}+3x^{3-1}$$

Subtract 1 from 2.

$$-8x^{1}+x^{1-1}+3x^{3-1}$$

Subtract 1 from 1.

$$-8x^{1}+x^{0}+3x^{3-1}$$

Subtract 1 from 3.

$$-8x^{1}+x^{0}+3x^{2}$$

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

$$-8x+x^{0}+3x^{2}$$

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

$$-8x+1+3x^{2}$$