What is the derivative of arctan(5x)?
Understand the Problem
The question is asking for the derivative of the function arctan(5x). To solve this, we will apply the chain rule of differentiation.
Answer
The derivative of $y = \arctan(5x)$ is given by $ \frac{5}{1 + 25x^2} $.
Answer for screen readers
The derivative of the function $y = \arctan(5x)$ is given by: $$ \frac{dy}{dx} = \frac{5}{1 + 25x^2} $$
Steps to Solve
- Identify the outer and inner functions
In this case, the outer function is $y = \arctan(u)$, and the inner function is $u = 5x$.
- Differentiate the outer function
The derivative of the outer function $y = \arctan(u)$ with respect to $u$ is given by the formula: $$ \frac{dy}{du} = \frac{1}{1 + u^2} $$
- Differentiate the inner function
Now differentiate the inner function $u = 5x$ with respect to $x$: $$ \frac{du}{dx} = 5 $$
- Apply the chain rule
Now we apply the chain rule to find the derivative of $y$ with respect to $x$: $$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$ Substituting the derivatives we found: $$ \frac{dy}{dx} = \frac{1}{1 + (5x)^2} \cdot 5 $$
- Simplify the result
Now simplify the expression: $$ \frac{dy}{dx} = \frac{5}{1 + 25x^2} $$
The derivative of the function $y = \arctan(5x)$ is given by: $$ \frac{dy}{dx} = \frac{5}{1 + 25x^2} $$
More Information
The derivative $ \frac{dy}{dx} $ tells us the rate of change of the function $ \arctan(5x) $ with respect to $ x $. The arctangent function is important in calculus and has applications in fields such as engineering and physics.
Tips
- Forgetting to apply the chain rule correctly can lead to incorrect results. Always ensure that both the inner and outer functions are properly differentiated.
