Solve 100-5x+2x^2-10x+x^2-30/2x+3x^2-25x+6x^2

$$100 - 5 x + 2 x ^ { 2 } - 10 x + x ^ { 2 } - \frac { 30 } { 2 } x + 3 x ^ { 2 } - 25 x + 6 x ^ { 2 }$$

Evaluate

$$12x^{2}-55x+100$$

Solution Steps

Combine -5x and -10x to get -15x.

$$100-15x+2x^{2}+x^{2}-\frac{30}{2}x+3x^{2}-25x+6x^{2}$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$100-15x+3x^{2}-\frac{30}{2}x+3x^{2}-25x+6x^{2}$$

Divide 30 by 2 to get 15.

$$100-15x+3x^{2}-15x+3x^{2}-25x+6x^{2}$$

Combine -15x and -15x to get -30x.

$$100-30x+3x^{2}+3x^{2}-25x+6x^{2}$$

Combine 3x^{2} and 3x^{2} to get 6x^{2}.

$$100-30x+6x^{2}-25x+6x^{2}$$

Combine -30x and -25x to get -55x.

$$100-55x+6x^{2}+6x^{2}$$

Combine 6x^{2} and 6x^{2} to get 12x^{2}.

$$100-55x+12x^{2}$$

Differentiate w.r.t. x

$$24x-55$$

Steps Using Derivative Rule for Sum

Combine -5x and -10x to get -15x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(100-15x+2x^{2}+x^{2}-\frac{30}{2}x+3x^{2}-25x+6x^{2})$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(100-15x+3x^{2}-\frac{30}{2}x+3x^{2}-25x+6x^{2})$$

Divide 30 by 2 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}x}(100-15x+3x^{2}-15x+3x^{2}-25x+6x^{2})$$

Combine -15x and -15x to get -30x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(100-30x+3x^{2}+3x^{2}-25x+6x^{2})$$

Combine 3x^{2} and 3x^{2} to get 6x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(100-30x+6x^{2}-25x+6x^{2})$$

Combine -30x and -25x to get -55x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(100-55x+6x^{2}+6x^{2})$$

Combine 6x^{2} and 6x^{2} to get 12x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(100-55x+12x^{2})$$

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}.

$$-55x^{1-1}+2\times 12x^{2-1}$$

Subtract 1 from 1.

$$-55x^{0}+2\times 12x^{2-1}$$

Multiply 2 times 12.

$$-55x^{0}+24x^{2-1}$$

Subtract 1 from 2.

$$-55x^{0}+24x^{1}$$

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

$$-55x^{0}+24x$$

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

$$-55+24x$$