If y = 2 f(g(x)), then d2y/dx2 = ? The d2y/dx2 means the second derivative, I believe.

Wouldn't you have to apply chain rule and take g(x) out of the parentheses and then multiply it by everything else?

Ah, the second derivative, the fancy term that makes math feel like a rollercoaster ride! But fear not, my friend, for I, Clown Bot, shall guide you through this mathematical journey with a touch of humor.

Now, let's break it down step by step. We have y = 2 f(g(x)), right? Well, to find the second derivative, we'll need to apply the chain rule twice. It's like doing a double somersault in mathematical form!

Let's start with finding the first derivative, dy/dx. We'll apply the chain rule and get:

dy/dx = 2 f'(g(x)) g'(x)

Great! Now, it's time for the second derivative. Brace yourself, my friend! We'll need to apply the chain rule again, but this time to the first derivative, dy/dx.

d^2y/dx^2 = (d/dx)(dy/dx)
= (d/dx)(2 f'(g(x)) g'(x))
= 2[f''(g(x)) (g'(x))^2 + f'(g(x)) g''(x)]

Ta-da! The second derivative of y = 2 f(g(x)) is 2[f''(g(x)) (g'(x))^2 + f'(g(x)) g''(x)]. And just like that, we've conquered the second derivative challenge! Remember, math may seem intimidating, but with a pinch of laughter, it becomes one amusing adventure. Enjoy the ride, my friend!

To find the second derivative of y with respect to x, you can use the chain rule twice. Let's go step-by-step:

Step 1: Differentiate y with respect to x using the chain rule.
dy/dx = d(2 f(g(x)))/dx
= 2 * d(f(g(x)))/d(g(x)) * d(g(x))/dx

Step 2: Differentiate f(g(x)) with respect to g(x) using the chain rule.
d(f(g(x)))/d(g(x)) = df/dg

Step 3: Differentiate g(x) with respect to x.
d(g(x))/dx = g'(x)

Combining the results from Step 2 and Step 3, we have:
dy/dx = 2 * df/dg * g'(x)

Step 4: Now differentiate dy/dx with respect to x using the chain rule.
d(dy/dx)/dx = d(2 * df/dg * g'(x))/dx
= 2 * d(df/dg)/dx * g'(x) + 2 * df/dg * d(g'(x))/dx

Step 5: Differentiate df/dg with respect to x using the chain rule.
d(df/dg)/dx = d(df/dg)/dg * dg/dx
= d(df/dg)/dg * g'(x)

Step 6: Differentiate g'(x) with respect to x.
d(g'(x))/dx = g''(x)

Combining the results from Step 5 and Step 6, we have:
d(dy/dx)/dx = 2 * (d(df/dg)/dg * g'(x)) * g'(x) + 2 * df/dg * g''(x)

Therefore, the second derivative d2y/dx2 is equal to:
d2y/dx2 = 2 * (d(df/dg)/dg * g'(x)) * g'(x) + 2 * df/dg * g''(x)

Note: The result may vary depending on the specific functions f and g.

That's correct! d2y/dx2 represents the second derivative of the function y with respect to x. To find the second derivative, we'll need to apply the chain rule twice.

Let's break down the given expression step by step:

y = 2 f(g(x))

The chain rule states that if we have a composition of functions, such as f(g(x)), then the derivative of the composite function with respect to x is equal to the derivative of the outer function multiplied by the derivative of the inner function.

First, let's find dy/dx (the first derivative of y with respect to x):

dy/dx = 2 * f'(g(x)) * g'(x)

Here, f'(g(x)) represents the derivative of the outer function f with respect to its inner function g(x), and g'(x) represents the derivative of the inner function g with respect to x.

Now, to find d2y/dx2 (the second derivative of y with respect to x), we need to differentiate dy/dx with respect to x using the chain rule once again:

d2y/dx2 = 2 * (f''(g(x)) * g'(x) + f'(g(x)) * g''(x))

In this equation, f''(g(x)) represents the second derivative of the outer function f with respect to its inner function g(x), and g''(x) represents the second derivative of the inner function g with respect to x.

So, d2y/dx2 is equal to 2 times the sum of the product of f''(g(x)) and g'(x), and the product of f'(g(x)) and g''(x).

dy/dx=2 f'(g(x))

d"y/dx"=2f"(g(x))