Solve x^{4}+3x^{3}-2x^3-5x^2-x+x^2+x+1+7x^4

$$x ^ { 4 } + 3 x ^ { 3 } - 2 x ^ { 3 } - 5 x ^ { 2 } - x + x ^ { 2 } + x + 1 + 7 x ^ { 4 }$$

Evaluate

$$8x^{4}+x^{3}-4x^{2}+1$$

Solution Steps

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

$$x^{4}+x^{3}-5x^{2}-x+x^{2}+x+1+7x^{4}$$

Combine -5x^{2} and x^{2} to get -4x^{2}.

$$x^{4}+x^{3}-4x^{2}-x+x+1+7x^{4}$$

Combine -x and x to get 0.

$$x^{4}+x^{3}-4x^{2}+1+7x^{4}$$

Combine x^{4} and 7x^{4} to get 8x^{4}.

$$8x^{4}+x^{3}-4x^{2}+1$$

Differentiate w.r.t. x

$$x\left(32x^{2}+3x-8\right)$$

Steps Using Derivative Rule for Sum

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}+x^{3}-5x^{2}-x+x^{2}+x+1+7x^{4})$$

Combine -5x^{2} and x^{2} to get -4x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}+x^{3}-4x^{2}-x+x+1+7x^{4})$$

Combine -x and x to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}+x^{3}-4x^{2}+1+7x^{4})$$

Combine x^{4} and 7x^{4} to get 8x^{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(8x^{4}+x^{3}-4x^{2}+1)$$

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}.

$$4\times 8x^{4-1}+3x^{3-1}+2\left(-4\right)x^{2-1}$$

Multiply 4 times 8.

$$32x^{4-1}+3x^{3-1}+2\left(-4\right)x^{2-1}$$

Subtract 1 from 4.

$$32x^{3}+3x^{3-1}+2\left(-4\right)x^{2-1}$$

Subtract 1 from 3.

$$32x^{3}+3x^{2}+2\left(-4\right)x^{2-1}$$

Multiply 3 times 1.

$$32x^{3}+3x^{2}-8x^{2-1}$$

Subtract 1 from 2.

$$32x^{3}+3x^{2}-8x^{1}$$

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

$$32x^{3}+3x^{2}-8x$$