Solve (2+x^3)^-1

$$( 2 + x ^ { 3 } ) ^ { - 1 }$$

Differentiate w.r.t. x

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

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).

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

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}.

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

Simplify.

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

Evaluate

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