$$ y=f(\{u\}) $$ Find derivative $$ \frac{d y}{d x} $$ $$ y=\frac{1}{(1+ an x)^{2}} $$

Question

$$ y=f(\{u\}) $$ Find derivative $$ \frac{d y}{d x} $$ $$ y=\frac{1}{(1+	an x)^{2}} $$

UpStudy AI · Accepted Answer

To find the derivative $$ \frac{dy}{dx} $$ for the function $$ y = \frac{1}{(1 + \tan x)^2} $$, we can use the chain rule and the quotient rule.

First, let's rewrite the function for clarity:

$$
y = (1 + \tan x)^{-2}
$$

Now, we will differentiate $$ y $$ with respect to $$ x $$:

1. **Differentiate using the chain rule**:
   $$
   \frac{dy}{dx} = -2(1 + \tan x)^{-3} \cdot \frac{d}{dx}(1 + \tan x)
   $$

2. **Differentiate $$ 1 + \tan x $$**:
   $$
   \frac{d}{dx}(1 + \tan x) = \sec^2 x
   $$

3. **Substituting back into the derivative**:
   $$
   \frac{dy}{dx} = -2(1 + \tan x)^{-3} \cdot \sec^2 x
   $$

4. **Rewriting the expression**:
   $$
   \frac{dy}{dx} = -\frac{2 \sec^2 x}{(1 + \tan x)^3}
   $$

Thus, the derivative $$ \frac{dy}{dx} $$ is:

$$
\frac{dy}{dx} = -\frac{2 \sec^2 x}{(1 + \tan x)^3}
$$
The derivative $$ \frac{dy}{dx} $$ is $$ -\frac{2 \sec^2 x}{(1 + \tan x)^3} $$.

\( y=f(\{u\}) \) Find derivative \( \frac{d y}{d x} \) \( y=\frac{1}{(1+\tan x)^{2}} \)