Differentiate w.r.t x
`tan (x^2)`

Answer

To differentiate the function \( y = \tan(x^2) \) with respect to \( x \), we will use the chain rule. Here are the steps to solve the problem:

### Step 1: Identify the outer and inner functions
Let \( y = \tan(u) \) where \( u = x^2 \). In this case, \( \tan(u) \) is the outer function and \( u = x^2 \) is the inner function.

### Step 2: Differentiate the outer function
The derivative of \( \tan(u) \) with respect to \( u \) is:
\[ \frac{dy}{du} = \sec^2(u) \]

### Step 3: Differentiate the inner function
Now, we differentiate the inner function \( u = x^2 \) with respect to \( x \):
\[ \frac{du}{dx} = 2x \]

### Step 4: Apply the chain rule
Using the chain rule, we can find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \sec^2(u) \cdot 2x \]

### Step 5: Substitute back the inner function
Now, substitute \( u = x^2 \) back into the equation:
\[ \frac{dy}{dx} = \sec^2(x^2) \cdot 2x \]

### Final Result
Thus, the derivative of \( y = \tan(x^2) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 2x \sec^2(x^2) \]

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