Solve d/dxleft({left(x^4/x^2+1right)}^3right)

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

Evaluate

$$\frac{6\left(x^{2}+2\right)x^{11}}{\left(x^{2}+1\right)^{4}}$$

Steps Using Chain Rule

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

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

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

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

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}.

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

Multiply x^{2}+1 times 4x^{4-1}.

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

Multiply x^{4} times 2x^{2-1}.

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

Simplify.

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

Differentiate w.r.t. x

$$\frac{6\left(5x^{4}+19x^{2}+22\right)x^{10}}{\left(x^{2}+1\right)^{5}}$$