Question

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$y=\frac{1}{\sqrt{x+2}}$$

Short answer

The derivative of the function $y=\frac{1}{\sqrt{x+2}}$ is $dy/dx=-1/(2\sqrt{(x+2{)}^3})$. The chain rule and the power rule were used in this differentiation process.

Step-by-step solution

Rewrite the function

To simplify the differentiation process, rewrite the function from a division as a multiplication using negative exponent. Rewrite the function $y=\frac{1}{\sqrt{x+2}}$ as $y=(x+2{)}^{-1/2}$.

Identify Outer and Inner Functions

In $y=(x+2{)}^{-1/2}$, the outer function is $f(x)=x^{-1/2}$ and the inner function is $g(x)=x+2$.

Apply the Chain Rule

The chain rule states that $dy/dx=dy/du\cdot du/dx$, where $u$ is the inner function $g(x)$. Therefore, applying the chain rule, $dy/dx=f^{\prime }(g(x))\cdot g^{\prime }(x)$.

Differentiate the Outer Function

For the outer function $f(x)=x^{-1/2}$, differentiate using the power rule: $f^{\prime }(x)=-1/2\cdot x^{-3/2}=-1/(2\sqrt{x^3})$. Place $g(x)=x+2$ into the derivative to get $f^{\prime }(g(x))=-1/(2\sqrt{(x+2{)}^3})$.

Differentiate the Inner Function

The derivative of $g(x)=x+2$ is $g^{\prime }(x)=1$.

Multiply the derivatives

Multiply $f^{\prime }(g(x))$ and $g^{\prime }(x)$ to find the derivative of the original function: $dy/dx=f^{\prime }(g(x))\cdot g^{\prime }(x)=-1/(2\sqrt{(x+2{)}^3})\cdot 1=-1/(2\sqrt{(x+2{)}^3})$.