Solve d/dx(24(x+x^3-x^5-x^3)/(1+x^2)6)

$$\frac { d } { d x } ( \frac { 24 ( x + x ^ { 3 } - x ^ { 5 } - x ^ { 3 } ) } { ( 1 + x ^ { 2 } ) 6 } )$$

Evaluate

$$4-12x^{2}$$

Steps Using Derivative Rule for Sum

Cancel out 6 in both numerator and denominator.

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

Combine -x^{3} and x^{3} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4\left(-x^{5}+x\right)}{x^{2}+1})$$

Factor the expressions that are not already factored in \frac{4\left(-x^{5}+x\right)}{x^{2}+1}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x\left(x-1\right)\left(x+1\right)\left(-x^{2}-1\right)}{x^{2}+1})$$

Extract the negative sign in -1-x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-4x\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)}{x^{2}+1})$$

Cancel out x^{2}+1 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-4x\left(x-1\right)\left(x+1\right))$$

Expand the expression.

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

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}.

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

Multiply 3 times -4.

$$-12x^{3-1}+4x^{1-1}$$

Subtract 1 from 3.

$$-12x^{2}+4x^{1-1}$$

Subtract 1 from 1.

$$-12x^{2}+4x^{0}$$

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

$$-12x^{2}+4\times 1$$

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

$$-12x^{2}+4$$

Differentiate w.r.t. x

$$-24x$$