Find the Difference Quotient f(x)=-5 square root of x+3

Solution:

Step 1:

Start with the difference quotient formula: $\frac{f(x+h)-f(x)}{h}$

Step 2:

Determine the function values needed for the formula.

Step 2.1:

Calculate $f(x+h)$ by substituting $x$ with $x+h$ in the function: $f(x+h)=-5\sqrt{(x+h)+3}$

Step 2.1.1:

Expand the expression to simplify: $f(x+h)=-5\sqrt{x+h+3}$

Step 2.1.2:

This simplification yields the expression: $-5\sqrt{x+h+3}$

Step 2.2:

Determine $f(x)$ and $f(x+h)$ using the given function: $f(x+h)=-5\sqrt{x+h+3}$ and $f(x)=-5\sqrt{x+3}$

Step 3:

Insert the calculated function values into the difference quotient formula: $\frac{f(x+h)-f(x)}{h}=\frac{-5\sqrt{x+h+3}-(-5\sqrt{x+3})}{h}$

Step 4:

Proceed to simplify the expression.

Step 4.1:

Focus on simplifying the numerator first.

Step 4.1.1:

Extract a factor of $-1$ from the terms: $\frac{-5\sqrt{x+h+3}-(-5\sqrt{x+3})}{h}$

Step 4.1.2:

Extract a common factor of $5$ from the terms: $\frac{-5(\sqrt{x+h+3}-\sqrt{x+3})}{h}$

Step 4.1.3:

Combine the multiplication of $-1$ and $5$: $\frac{-5(\sqrt{x+h+3}-\sqrt{x+3})}{h}$

Step 4.2:

Position the negative sign in front of the fraction: $-\frac{5(\sqrt{x+h+3}-\sqrt{x+3})}{h}$

Step 5:

The process is complete, and the difference quotient is simplified.

Knowledge Notes:

The difference quotient is a formula used in calculus to determine the slope of the secant line between two points on a graph of a function. It is given by $\frac{f(x+h)-f(x)}{h}$, where $f(x)$ is the function value at $x$, $f(x+h)$ is the function value at $x+h$, and $h$ is the difference between the two x-values.

To find the difference quotient for a given function, you need to:

1. Evaluate the function at $x+h$ and at $x$.

2. Substitute these values into the difference quotient formula.

3. Simplify the resulting expression, if possible.

In this problem, the function given is $f(x)=-5\sqrt{x+3}$. When finding the difference quotient, we substitute $x$ with $x+h$ in the function to get $f(x+h)$. We then simplify the expression by combining like terms and factoring out common factors. The goal is to simplify the expression as much as possible, which often involves combining like terms, factoring, and reducing fractions.