Question

Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ y=\frac{1}{\sqrt{3 x+5}} $$

Short answer

The derivative of the function \( y=\frac{1}{\sqrt{3 x+5}} \) is \( y'= -\frac{3}{2(3x+5)\sqrt{3x+5}} \)

Step-by-step solution

Application of Quotient Rule

The general form of the Quotient Rule is \(\frac{d}{dx}(\frac{u}{v}) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}\). In the given function, \( u = 1 \) and \( v = \sqrt{3x+5} \).

Differentiation of u and v

The derivative of u is zero because u is a constant. As for v, the chain rule is applied. The chain rule is \( \frac{d}{dx}[f(g(x))] = f^\')(g(x)) \cdot g\)'(x) \). Here, \( f(x) = \sqrt{x} \) and \( g(x) = 3x+5 \). So, \( f\)'(x) is \(\frac{1}{2\sqrt{x}}\), \( g\)'(x) is 3. Therefore, \( v\)' equals \( \frac{1}{2\sqrt{3x+5}} \) times 3.

Substitute and Simplify

Then substitute u, v, \( u\)' and \( v\)' back into the quotient rule:\[ y'= \frac{\sqrt{3x+5}(0)-1(\frac{1}{2\sqrt{3x+5}} \times 3}{(\sqrt{3x+5})^2} = -\frac{3}{2(3x+5)\sqrt{3x+5}} \]This is the derivative of the function.