Questions: If f(x) = (3 x^2 tan x) / (sec x) find f'(x). Find f'(1).

Solution Steps

To find the derivative \( f'(x) \) of the function \( f(x) = \frac{3x^2 \tan x}{\sec x} \), we can simplify the function first and then apply the quotient rule or product rule as needed. After finding the derivative, we can substitute \( x = 1 \) to find \( f'(1) \).

We start with the function \[ f(x) = \frac{3x^2 \tan x}{\sec x}. \] Using the identity \(\sec x = \frac{1}{\cos x}\), we can rewrite the function as \[ f(x) = 3x^2 \tan x \cdot \cos x = 3x^2 \sin x. \]

Next, we differentiate \(f(x)\) with respect to \(x\): \[ f'(x) = \frac{d}{dx}(3x^2 \sin x). \] Using the product rule, we find: \[ f'(x) = 3x^2 \cos x + 6x \sin x. \]

Now, we substitute \(x = 1\) into the derivative: \[ f'(1) = 3 \cos(1) + 6 \sin(1). \] Calculating the values, we have: \[ f'(1) \approx 3 \cdot 0.5403 + 6 \cdot 0.8415 \approx 1.6209 + 5.0490 \approx 6.6699. \]

Final Answer