Solve (1+x^2)^frac{1{3}}

$$( 1 + x ^ { 2 } ) ^ { \frac { 1 } { 3 } }$$

Differentiate w.r.t. x

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

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).

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

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}.

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

Simplify.

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

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

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

Evaluate

$$\sqrt[3]{x^{2}+1}$$