Find f’’(x) if f(x)=(6^x^2+3)

To find the second derivative of the function f(x) = (6^(x^2) + 3), we will need to differentiate it twice.

The first step is to find the derivative of f(x) with respect to x, denoted as f'(x). The derivative of (6^(x^2) + 3) can be calculated by applying the chain rule.

Step 1: Find f'(x)
To find f'(x), we differentiate each term separately. Let's start by differentiating the first term 6^(x^2) and then the constant term 3:

- Differentiating the term 6^(x^2) involves using the chain rule. The general formula for the chain rule is: (d/dx) [f(g(x))] = f'(g(x)) * g'(x).
- Here, f(g(x)) = 6^x and g(x) = x^2.
- Applying the chain rule, the derivative of 6^(x^2) is: (d/dx) [6^(x^2)] = ln(6) * 6^(x^2) * 2x, where ln(6) is the natural logarithm of 6.

- The constant term 3 differentiates to 0 since its derivative is always 0.

So, f'(x) = ln(6) * 6^(x^2) * 2x + 0
Simplifying, f'(x) = 2x ln(6) * 6^(x^2).

Now that we have f'(x), let's find the second derivative f''(x).

Step 2: Find f''(x)
To find f''(x), we differentiate f'(x) with respect to x:

- Differentiating 2x ln(6) * 6^(x^2) involves applying the product rule: (d/dx) [f(x) * g(x)] = f'(x) * g(x) + f(x) * g'(x).

- The first term 2x differentiates to 2, and the second term ln(6) * 6^(x^2) remains unchanged.

- Differentiating 6^(x^2) * 2x involves using the chain rule once again.

- Applying the chain rule, the derivative of 6^(x^2) * 2x is: ln(6) * 6^(x^2) * 2x * 2x + 2 * 6^(x^2).

Combining all the terms, we get:
f''(x) = 2 + ln(6) * 6^(x^2) * 4x^2 + 2 * 6^(x^2).
Simplifying, f''(x) = 2 + 2 * 6^(x^2) + 4x^2 * ln(6) * 6^(x^2).

Therefore, the second derivative of f(x) = (6^(x^2) + 3) is f''(x) = 2 + 2 * 6^(x^2) + 4x^2 * ln(6) * 6^(x^2).