Evaluate (x^2 - 25) / (x^2 + 25) when x = 2 and determine dy/dx.
Understand the Problem
The question asks to evaluate the expression (x^2 - 25) / (x^2 + 25) when x = 2 and to find the derivative dy/dx of the same expression. These are two separate tasks: first, substitute x = 2 into the expression; second, find the derivative of the expression with respect to x.
Answer
Value at $x=2$: $-\frac{21}{29}$. Derivative: $\frac{100x}{(x^2 + 25)^2}$.
Answer for screen readers
Value of the expression $\frac{x^2 - 25}{x^2 + 25}$ at $x=2$ is $-\frac{21}{29}$.
The derivative $\frac{dy}{dx}$ of the expression $\frac{x^2 - 25}{x^2 + 25}$ is $\frac{100x}{(x^2 + 25)^2}$.
Steps to Solve
- Evaluate the expression at $x=2$
Substitute $x = 2$ into the expression $\frac{x^2 - 25}{x^2 + 25}$:
$\frac{(2)^2 - 25}{(2)^2 + 25} = \frac{4 - 25}{4 + 25} = \frac{-21}{29}$
- Find the derivative using the quotient rule
The quotient rule states that if $y = \frac{u}{v}$, then $\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$. In this case, $u = x^2 - 25$ and $v = x^2 + 25$.
- Compute $\frac{du}{dx}$
$\frac{du}{dx} = \frac{d}{dx}(x^2 - 25) = 2x$
- Compute $\frac{dv}{dx}$
$\frac{dv}{dx} = \frac{d}{dx}(x^2 + 25) = 2x$
- Apply the quotient rule
$\frac{dy}{dx} = \frac{(x^2 + 25)(2x) - (x^2 - 25)(2x)}{(x^2 + 25)^2}$
- Simplify the expression
$\frac{dy}{dx} = \frac{2x^3 + 50x - (2x^3 - 50x)}{(x^2 + 25)^2} = \frac{2x^3 + 50x - 2x^3 + 50x}{(x^2 + 25)^2} = \frac{100x}{(x^2 + 25)^2}$
Value of the expression $\frac{x^2 - 25}{x^2 + 25}$ at $x=2$ is $-\frac{21}{29}$.
The derivative $\frac{dy}{dx}$ of the expression $\frac{x^2 - 25}{x^2 + 25}$ is $\frac{100x}{(x^2 + 25)^2}$.
More Information
The quotient rule is a fundamental concept in calculus and is used to find the derivative of a function that is the ratio of two other functions.
Tips
A common mistake is incorrectly applying the quotient rule, especially getting the order of terms in the numerator wrong. Remembering the formula $\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$ can help avoid this mistake. Another common mistake is making errors in the algebraic simplification after applying the quotient rule. Care should be taken to distribute and combine like terms correctly.
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