Solve (x^{4}-6x^{5}+8x^2-5)+(-8x^4+3-7x^2+10x^2-5)

$$( x ^ { 4 } - 6 x ^ { 5 } + 8 x ^ { 2 } - 5 ) + ( - 8 x ^ { 4 } + 3 - 7 x ^ { 2 } + 10 x ^ { 2 } - 5 )$$

Evaluate

$$-6x^{5}-7x^{4}+11x^{2}-7$$

Solution Steps

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

$$-7x^{4}-6x^{5}+8x^{2}-5+3-7x^{2}+10x^{2}-5$$

Add -5 and 3 to get -2.

$$-7x^{4}-6x^{5}+8x^{2}-2-7x^{2}+10x^{2}-5$$

Combine 8x^{2} and -7x^{2} to get x^{2}.

$$-7x^{4}-6x^{5}+x^{2}-2+10x^{2}-5$$

Combine x^{2} and 10x^{2} to get 11x^{2}.

$$-7x^{4}-6x^{5}+11x^{2}-2-5$$

Subtract 5 from -2 to get -7.

$$-7x^{4}-6x^{5}+11x^{2}-7$$

Differentiate w.r.t. x

$$22x-28x^{3}-30x^{4}$$

Steps Using Derivative Rule for Sum

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(-7x^{4}-6x^{5}+8x^{2}-5+3-7x^{2}+10x^{2}-5)$$

Add -5 and 3 to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-7x^{4}-6x^{5}+8x^{2}-2-7x^{2}+10x^{2}-5)$$

Combine 8x^{2} and -7x^{2} to get x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-7x^{4}-6x^{5}+x^{2}-2+10x^{2}-5)$$

Combine x^{2} and 10x^{2} to get 11x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-7x^{4}-6x^{5}+11x^{2}-2-5)$$

Subtract 5 from -2 to get -7.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-7x^{4}-6x^{5}+11x^{2}-7)$$

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}.

$$4\left(-7\right)x^{4-1}+5\left(-6\right)x^{5-1}+2\times 11x^{2-1}$$

Multiply 4 times -7.

$$-28x^{4-1}+5\left(-6\right)x^{5-1}+2\times 11x^{2-1}$$

Subtract 1 from 4.

$$-28x^{3}+5\left(-6\right)x^{5-1}+2\times 11x^{2-1}$$

Multiply 5 times -6.

$$-28x^{3}-30x^{5-1}+2\times 11x^{2-1}$$

Subtract 1 from 5.

$$-28x^{3}-30x^{4}+2\times 11x^{2-1}$$

Multiply 5 times -6.

$$-28x^{3}-30x^{4}+22x^{2-1}$$

Subtract 1 from 2.

$$-28x^{3}-30x^{4}+22x^{1}$$

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

$$-28x^{3}-30x^{4}+22x$$