Solve R(x)=1/x^2+4x+4

$$R ( x ) = \frac { 1 } { x ^ { 2 } + 4 x + 4 }$$

Differentiate w.r.t. x

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

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).

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

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}.

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

Simplify.

$$\left(x^{2}+4x^{1}+4\right)^{-2}\left(-2x^{1}-4x^{0}\right)$$

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

$$\left(x^{2}+4x+4\right)^{-2}\left(-2x-4x^{0}\right)$$

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

$$\left(x^{2}+4x+4\right)^{-2}\left(-2x-4\right)$$

Evaluate

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