https://file.aiquickdraw.com/mathbot/file/1728093427184tieqg7tw.jpg

The given function is \( y = 3 \tan x \), and we are asked to find the second derivative \( y'' \).

### Step 1: Find the first derivative \( y' \)
To find the first derivative of \( y = 3 \tan x \), we use the derivative of \( \tan x \), which is \( \sec^2 x \).

\[
y' = \frac{d}{dx} (3 \tan x) = 3 \sec^2 x
\]

### Step 2: Find the second derivative \( y'' \)
Now, we need to differentiate \( y' = 3 \sec^2 x \).

Using the chain rule:
\[
\frac{d}{dx} (\sec^2 x) = 2 \sec^2 x \cdot \sec x \tan x = 2 \sec^3 x \tan x
\]

Thus, the second derivative is:
\[
y'' = \frac{d}{dx} (3 \sec^2 x) = 3 \cdot 2 \sec^3 x \tan x = 6 \sec^3 x \tan x
\]

### Final Answer:
\[
y'' = 6 \sec^3 x \tan x
\]

Would you like a more detailed explanation of any step or have any questions?

Here are five related questions:
1. What is the derivative of \( \tan^{-1}(x) \)?
2. How do you find the derivative of trigonometric functions like \( \sin(x) \) and \( \cos(x) \)?
3. Can you explain the chain rule in differentiation?
4. What is the geometric meaning of the second derivative?
5. How does the secant function behave for large values of \( x \)?

**Tip:** Remember, when differentiating trigonometric functions, it's essential to know their basic derivatives and how to apply the chain rule efficiently.

Learn how to find the second derivative of y = 3 tan x using the chain rule and trigonometric differentiation. This solution walks through the steps, starting from y' = 3 sec^2 x to y'' = 6 sec^3 x tan x. Suitable for advanced high school calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation