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.

Key concepts

Quotient Rule

The quotient rule is a key part of calculus used to find the derivative of two divided functions. When you have a function \ y = \frac{u}{v} \, where both \ u \ and \ v \ are differentiable, the rule helps to find the derivative systematically. The quotient rule formula is given by:
\[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]
This formula works by taking the derivative of the numerator and the denominator separately, and then piecing them together. Applying the quotient rule helps in situations where functions are deeply intertwined in division.

• Step 1: Differentiate the top function \(u\).
• Step 2: Differentiate the bottom function \(v\).
• Step 3: Substitute these derivatives into the formula.
• Step 4: Simplify as much as possible.

By using this, you can tackle complex divisions methodically.

Chain Rule

The chain rule is essential in calculus when dealing with composite functions. A composite function is one that involves another function inside it. If you have\[ y = f(g(x)) \], the chain rule helps you find the derivative. The principle of the chain rule is:
\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \]
This means you differentiate the outer function\( f \) with the inner function still inside, and multiply by the derivative of the inner function \( g \).
For instance, with\( f(x) = \sqrt{x} \) and\( g(x) = 3x + 5 \), you find\( f'(x) = \frac{1}{2\sqrt{x}} \) and\( g'(x) = 3 \).
This method is particularly helpful for functions within functions, ensuring you capture all the layers.

Differentiation

Differentiation is the process of finding a derivative, representing the rate of change of a function. It expresses how a small change in one variable affects another variable dependent on it.

• Basic Idea: It's like finding the slope of a curve at any given point.
• Powers, Products, and Quotients: Rules like power rule, product rule, and quotient rule guide differentiation.
• Chains: For nested functions, the chain rule is essential.

Through differentiation, you're determining the exact change or slope at any segment of a curve. It provides insights into the function's behavior, allowing predictions and deeper analysis.

Calculus

Calculus is a branch of mathematics focusing on change and motion, foundational to many scientific fields. It is divided into two main parts: differential calculus and integral calculus. Differential calculus deals with the concept of a derivative, essential for understanding rates of change.

• Core Concepts: Limits, derivatives, and integration.
• Applications: Used in physics, engineering, economics, and statistics.
• Problem Solving: Helps solve complex problems about motion, areas, and rates.

Understanding calculus enables tackling both theoretical and real-world problems where change is a factor. It requires grasping not only the methods but also the reasons behind their application, providing a robust mathematical toolset for exploring how things change systematically.