Find dy/dx y=((1+x^2)/(1-x^2))^17

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 $f^{\prime }(g(x))g^{\prime }(x)$. 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 $17u^{16}$:

$$17u^{16}\cdot \frac{d}{dx}\left(\frac{1+x^2}{1-x^2}\right)$$

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:

1. 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 $\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$.

2. Power Rule: A basic differentiation rule that states if $y=x^n$, then $\frac{dy}{dx}=n\cdot x^{n-1}$.

3. Quotient Rule: A rule for differentiating functions in the form of a quotient. If $y=\frac{f(x)}{g(x)}$, then $\frac{dy}{dx}=\frac{g(x)\cdot f^{\prime }(x)-f(x)\cdot g^{\prime }(x)}{[g(x){]}^2}$.

4. 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 $\frac{dy}{dx}=\frac{df(x)}{dx}+\frac{dg(x)}{dx}$.

5. 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.