Solve d/dx(x^2+4x)^frac{1{2}}

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

Evaluate

$$\frac{x+2}{\sqrt{x\left(x+4\right)}}$$

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).

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

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}.

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

Simplify.

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

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

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

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

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

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

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

Differentiate w.r.t. x

$$-\frac{4}{\left(x\left(x+4\right)\right)^{\frac{3}{2}}}$$