Solve (x^4-6x^4+18x^4-34x+21)+(x^2-2x+7)

$$( x ^ { 4 } - 6 x ^ { 4 } + 18 x ^ { 4 } - 34 x + 21 ) + ( x ^ { 2 } - 2 x + 7 )$$

Evaluate

$$13x^{4}+x^{2}-36x+28$$

Solution Steps

Combine x^{4} and -6x^{4} to get -5x^{4}.

$$-5x^{4}+18x^{4}-34x+21+x^{2}-2x+7$$

Combine -5x^{4} and 18x^{4} to get 13x^{4}.

$$13x^{4}-34x+21+x^{2}-2x+7$$

Combine -34x and -2x to get -36x.

$$13x^{4}-36x+21+x^{2}+7$$

Add 21 and 7 to get 28.

$$13x^{4}-36x+28+x^{2}$$

Differentiate w.r.t. x

$$2\left(26x^{3}+x-18\right)$$

Steps Using Derivative Rule for Sum

Combine x^{4} and -6x^{4} to get -5x^{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-5x^{4}+18x^{4}-34x+21+x^{2}-2x+7)$$

Combine -5x^{4} and 18x^{4} to get 13x^{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(13x^{4}-34x+21+x^{2}-2x+7)$$

Combine -34x and -2x to get -36x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(13x^{4}-36x+21+x^{2}+7)$$

Add 21 and 7 to get 28.

$$\frac{\mathrm{d}}{\mathrm{d}x}(13x^{4}-36x+28+x^{2})$$

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}.

$$4\times 13x^{4-1}-36x^{1-1}+2x^{2-1}$$

Multiply 4 times 13.

$$52x^{4-1}-36x^{1-1}+2x^{2-1}$$

Subtract 1 from 4.

$$52x^{3}-36x^{1-1}+2x^{2-1}$$

Subtract 1 from 1.

$$52x^{3}-36x^{0}+2x^{2-1}$$

Subtract 1 from 2.

$$52x^{3}-36x^{0}+2x^{1}$$

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

$$52x^{3}-36x^{0}+2x$$

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

$$52x^{3}-36+2x$$