Differentiate y = [(-tanx)tanx]
Understand the Problem
The question is asking us to find the derivative of $y$ with respect to $x$, where $y = [(-tanx)tanx]$. We need to apply differentiation rules to find the derivative of given function.
Answer
$\frac{dy}{dx} = -2tan(x)sec^2x$
Answer for screen readers
$\frac{dy}{dx} = -2tan(x)sec^2x$
Steps to Solve
- Simplify the function
First, we simplify the given function $y = [(-tanx)tanx]$. This can be rewritten as $y = -tan^2x$.
- Differentiate using the chain rule
To find the derivative of $y$ with respect to $x$, we need to differentiate $y = -tan^2x$. We'll use the chain rule. Let $u = tan(x)$, then $y = -u^2$. The derivative of $y$ with respect to $u$ is: $\frac{dy}{du} = -2u$ And the derivative of $u$ with respect to $x$ is: $\frac{du}{dx} = sec^2x$
- Apply the chain rule formula
Using the chain rule, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. Substitute the derivatives we found in the previous step:
$\frac{dy}{dx} = -2u \cdot sec^2x$
- Substitute $u = tan(x)$
Substitute $u = tan(x)$ back into the equation:
$\frac{dy}{dx} = -2tan(x)sec^2x$
$\frac{dy}{dx} = -2tan(x)sec^2x$
More Information
The derivative of $y = -tan^2(x)$ is $-2tan(x)sec^2(x)$. This tells us how the value of $y$ changes with respect to changes in $x$.
Tips
A common mistake is forgetting to apply the chain rule correctly. Another mistake could be an incorrect derivative of $tan(x)$.
AI-generated content may contain errors. Please verify critical information
