Solve |x^2+1|x(x^2-5)*|x-2|

$$| x ^ { 2 } + 1 | x ( x ^ { 2 } - 5 ) \times | x - 2 |$$

Differentiate w.r.t. x

$$sign(x-2)x^{5}+5|x-2|x^{4}-4sign(x-2)x^{3}-12|x-2|x^{2}-5xsign(x-2)-5|x-2|$$

Steps Using Derivative Rule for Sum

Use the distributive property to multiply |x^{2}+1|x by x^{2}-5.

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

Use the distributive property to multiply |x^{2}+1|x^{3}-5|x^{2}+1|x by |x-2|.

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

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}.

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

Subtract 1 from 3.

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

Subtract 1 from 1.

$$3|x-2|\left(x^{2}+1\right)x^{2}+\left(-5|x-2|\left(x^{2}+1\right)\right)x^{0}$$

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

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

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

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

Evaluate

$$x|x-2|\left(x^{2}-5\right)\left(x^{2}+1\right)$$