Find dy/dx y=(tan(x))/(1+tan(x))

Solution:

Step 1:

Take the derivative of both sides with respect to $x$: $\frac{d}{dx}(y)=\frac{d}{dx}\left(\frac{\tan (x)}{1+\tan (x)}\right)$

Step 2:

The derivative of $y$ with respect to $x$ is denoted as $\frac{dy}{dx}$.

Step 3:

Apply the derivative to the right-hand side.

Step 3.1:

Use the Quotient Rule: $\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$, where $u=\tan (x)$ and $v=1+\tan (x)$.

Step 3.2:

Find the derivative of $\tan (x)$, which is ${\sec }^2(x)$.

Step 3.3:

Proceed with differentiation.

Step 3.3.1:

Apply the Sum Rule to differentiate $1+\tan (x)$: $\frac{d}{dx}(1)+\frac{d}{dx}(\tan (x))$.

Step 3.3.2:

The derivative of a constant, $1$, is $0$.

Step 3.3.3:

Combine $0$ and the derivative of $\tan (x)$.

Step 3.4:

The derivative of $\tan (x)$ is ${\sec }^2(x)$.

Step 3.5:

Simplify the expression.

Step 3.5.1:

Distribute ${\sec }^2(x)$ across the terms.

Step 3.5.2:

Reduce the numerator.

Step 3.5.2.1:

Cancel out like terms.

Step 3.5.2.1.1:

Subtract $\tan (x){\sec }^2(x)$ from itself to get $0$.

Step 3.5.2.1.2:

Combine ${\sec }^2(x)$ and $0$.

Step 3.5.2.2:

Multiply ${\sec }^2(x)$ by $1$.

Step 4:

Express the left side as equal to the simplified right side: $y=\frac{{\sec }^2(x)}{(1+\tan (x){)}^2}$.

Step 5:

Substitute $\frac{dy}{dx}$ for $y$: $\frac{dy}{dx}=\frac{{\sec }^2(x)}{(1+\tan (x){)}^2}$.

Knowledge Notes:

1. Derivative: The derivative of a function measures how the function value changes as its input changes. The notation $\frac{d}{dx}$ is used to denote the derivative with respect to $x$.

2. Quotient Rule: A derivative rule used when differentiating a quotient of two functions. It states that $\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$.

3. Sum Rule: A derivative rule that allows us to take derivatives of functions that are sums. It states that the derivative of a sum is the sum of the derivatives.

4. Derivative of $\tan (x)$: The derivative of $\tan (x)$ with respect to $x$ is ${\sec }^2(x)$.

5. Constant Rule: The derivative of a constant is zero.

6. Simplification: The process of reducing an expression to its simplest form by performing algebraic operations and combining like terms.