Solve d/dxleft(1/x(x+y)right)

$$\frac{d}{d x } \left( \frac{ 1 }{ x(x+y) } \right)$$

Evaluate

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

Steps Using Chain Rule

Use the distributive property to multiply x by x+y.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{2}+xy})$$

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).

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

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}.

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

Simplify.

$$\left(x^{2}+yx^{1}\right)^{-2}\left(-2x^{1}+\left(-y\right)x^{0}\right)$$

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

$$\left(x^{2}+yx\right)^{-2}\left(-2x+\left(-y\right)x^{0}\right)$$

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

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

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

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

Differentiate w.r.t. x

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