Solve x^3-3x+1x^5+4x^3+x^2+3x+1

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

Evaluate

$$x^{5}+5x^{3}+x^{2}+1$$

Solution Steps

Combine x^{3} and 4x^{3} to get 5x^{3}.

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

Combine -3x and 3x to get 0.

$$5x^{3}+1x^{5}+x^{2}+1$$

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

$$5x^{3}+x^{5}+x^{2}+1$$

Differentiate w.r.t. x

$$x\left(5x^{3}+15x+2\right)$$

Steps Using Derivative Rule for Sum

Combine x^{3} and 4x^{3} to get 5x^{3}.

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

Combine -3x and 3x to get 0.

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

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}.

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

Multiply 3 times 5.

$$15x^{3-1}+5x^{5-1}+2x^{2-1}$$

Subtract 1 from 3.

$$15x^{2}+5x^{5-1}+2x^{2-1}$$

Subtract 1 from 5.

$$15x^{2}+5x^{4}+2x^{2-1}$$

Subtract 1 from 2.

$$15x^{2}+5x^{4}+2x^{1}$$

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

$$15x^{2}+5x^{4}+2x$$