Find dy/dx y=(tan(x))/(1+tan(x))
The problem presented is a calculus problem that involves finding the derivative of a given function with respect to x. The function y is defined as a quotient where the numerator is the tangent of x, denoted as tan(x), and the denominator is the expression 1 added to the tangent of x. The goal is to determine the rate at which y changes as x changes, which is calculated by finding the derivative of y with respect to x, commonly written as dy/dx. Calculating this derivative will involve applying differentiation rules for trigonometric functions and the quotient rule, which is the method used to differentiate ratios of functions.
$y = \frac{tan \left(\right. x \left.\right)}{1 + tan \left(\right. x \left.\right)}$
Answer
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:
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 $x$.
Quotient Rule: A derivative rule used when differentiating a quotient of two functions. It states that $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$.
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.
Derivative of $\tan(x)$: The derivative of $\tan(x)$ with respect to $x$ is $\sec^2(x)$.
Constant Rule: The derivative of a constant is zero.
Simplification: The process of reducing an expression to its simplest form by performing algebraic operations and combining like terms.
