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

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

Evaluate

$$5+11x^{2}-4x^{3}-5x^{4}$$

Solution Steps

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

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

Combine 4x^{2} and 7x^{2} to get 11x^{2}.

$$11x^{2}+5-4x^{3}-5x^{4}$$

Differentiate w.r.t. x

$$2x\left(11-6x-10x^{2}\right)$$

Steps Using Derivative Rule for Sum

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

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

Combine 4x^{2} and 7x^{2} to get 11x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(11x^{2}+5-4x^{3}-5x^{4})$$

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}.

$$2\times 11x^{2-1}+3\left(-4\right)x^{3-1}+4\left(-5\right)x^{4-1}$$

Multiply 2 times 11.

$$22x^{2-1}+3\left(-4\right)x^{3-1}+4\left(-5\right)x^{4-1}$$

Subtract 1 from 2.

$$22x^{1}+3\left(-4\right)x^{3-1}+4\left(-5\right)x^{4-1}$$

Multiply 3 times -4.

$$22x^{1}-12x^{3-1}+4\left(-5\right)x^{4-1}$$

Subtract 1 from 3.

$$22x^{1}-12x^{2}+4\left(-5\right)x^{4-1}$$

Multiply 3 times -4.

$$22x^{1}-12x^{2}-20x^{4-1}$$

Subtract 1 from 4.

$$22x^{1}-12x^{2}-20x^{3}$$

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

$$22x-12x^{2}-20x^{3}$$