Solve d/dx(1/sqrt{x+3})^2

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

Evaluate

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

Steps Using Chain Rule

To raise \frac{1}{\sqrt{x+3}} to a power, raise both numerator and denominator to the power and then divide.

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

Calculate 1 to the power of 2 and get 1.

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

Calculate \sqrt{x+3} to the power of 2 and get x+3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x+3})$$

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).

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

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}.

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

Simplify.

$$-x^{0}\left(x^{1}+3\right)^{-2}$$

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

$$-x^{0}\left(x+3\right)^{-2}$$

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

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

Differentiate w.r.t. x

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