Find dy/dx y=((1+x^2)/(1-x^2))^17
The question is asking for the derivative of the function y with respect to x, where y is a function of x defined as ((1+x^2)/(1-x^2))^17. The derivative, denoted as dy/dx, is a measure of how y changes as x changes. In this case, you would need to apply calculus techniques to find the rate of change of the function at any point along its curve. The technique likely to be used here involves the chain rule because the function is a composite function, raising a quotient to a power, and the quotient rule, because of the division of two functions of x within the exponentiation. The order of applying these rules is crucial to simplifying the expression correctly to get the derivative.
$y = \left(\left(\right. \frac{1 + x^{2}}{1 - x^{2}} \left.\right)\right)^{17}$
Answer
Solution:
Step 1
Take the derivative of both sides of the equation with respect to $x$:
$$\frac{d}{dx} y = \frac{d}{dx} \left(\frac{1 + x^2}{1 - x^2}\right)^{17}$$
Step 2
The derivative of $y$ with respect to $x$ is denoted as $\frac{dy}{dx}$.
Step 3
To differentiate the right-hand side, apply the chain rule.
Step 3.1
The chain rule states that the derivative of a composite function $f(g(x))$ is $f'(g(x))g'(x)$. Let $f(x) = x^{17}$ and $g(x) = \frac{1 + x^2}{1 - x^2}$.
Step 3.1.1
Introduce a substitution $u = \frac{1 + x^2}{1 - x^2}$ and differentiate:
$$\frac{d}{du} u^{17} \cdot \frac{d}{dx} \left(\frac{1 + x^2}{1 - x^2}\right)$$
Step 3.1.2
Apply the power rule to $u^{17}$ to get $17u^{16}$:
$$17 u^{16} \cdot \frac{d}{dx} \left(\frac{1 + x^2}{1 - x^2}\right)$$
Step 3.1.3
Substitute back $u$ with $\frac{1 + x^2}{1 - x^2}$:
$$17 \left(\frac{1 + x^2}{1 - x^2}\right)^{16} \cdot \frac{d}{dx} \left(\frac{1 + x^2}{1 - x^2}\right)$$
Step 3.2
Apply the quotient rule to differentiate $\frac{1 + x^2}{1 - x^2}$.
Step 3.3
Differentiate the numerator and the denominator separately.
Step 3.3.1
The derivative of $1 + x^2$ is the sum of the derivatives of $1$ and $x^2$.
Step 3.3.2
The derivative of a constant is zero, so the derivative of $1$ is $0$.
Step 3.3.3
Combine $0$ and the derivative of $x^2$.
Step 3.3.4
Apply the power rule to $x^2$ to get $2x$.
Step 3.3.5
Combine the terms involving $2x$.
Step 3.3.6
Differentiate $1 - x^2$ as the sum of the derivatives of $1$ and $-x^2$.
Step 3.3.7
The derivative of $1$ is zero.
Step 3.3.8
Combine $0$ and the derivative of $-x^2$.
Step 3.3.9
The derivative of $-x^2$ is $-2x$.
Step 3.3.10
Combine the terms involving $2x$.
Step 3.4
Simplify the expression by combining like terms and applying algebraic rules.
Step 4
Express the derivative $\frac{dy}{dx}$ as the simplified expression.
Step 5
Replace $y$ with $\frac{dy}{dx}$ to complete the differentiation.
Knowledge Notes:
Chain Rule: A fundamental differentiation rule used for finding the derivative of composite functions. It states that if $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.
Power Rule: A basic differentiation rule that states if $y = x^n$, then $\frac{dy}{dx} = n \cdot x^{n-1}$.
Quotient Rule: A rule for differentiating functions in the form of a quotient. If $y = \frac{f(x)}{g(x)}$, then $\frac{dy}{dx} = \frac{g(x) \cdot f'(x) - f(x) \cdot g'(x)}{[g(x)]^2}$.
Sum Rule: This rule states that the derivative of a sum of functions is the sum of the derivatives of those functions. If $y = f(x) + g(x)$, then $\frac{dy}{dx} = \frac{df(x)}{dx} + \frac{dg(x)}{dx}$.
Difference of Squares: An algebraic pattern that states $a^2 - b^2 = (a + b)(a - b)$, which is often used in simplifying expressions involving squares.
By applying these rules and properties systematically, we can differentiate complex functions step by step.
