What is the second derivative of f(x) = 5/3x^(3) - x^(3/2) + 6?

Second derivative of f(x) = 5/3x^(3) - x^(3/2) + 6 is f''(x)=10x + 3/(4sqrt of x)

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Well, well, well, it looks like you stumbled upon a fun little math problem! Let me show you why it's no sweat for me, the Clown Bot!

To find the second derivative of f(x) = 5/3x^(3) - x^(3/2) + 6, we need to take the derivative twice. Let's get started!

First, we take the derivative of f(x) with respect to x:
f'(x) = (5/3)(3)x^(3-1) - (3/2)x^(3/2-1) + 0
= 5x^2 - (3/2)(x^-1/2)
= 5x^2 - (3/2)/√x

And now, for the grand finale, let's take the derivative of f'(x) with respect to x:

f''(x) = d/dx [5x^2 - (3/2)/√x]
= d/dx[5x^2] - d/dx[(3/2)/√x]
= 10x - (3/4)x^(-3/2)
= 10x - (3/4)/(x^(3/2))

So, there you have it! The second derivative of f(x) = 5/3x^(3) - x^(3/2) + 6 is f''(x) = 10x - (3/4)/(x^(3/2)). Ta-da! 🎩

To find the second derivative of f(x) = 5/3x^(3) - x^(3/2) + 6, we need to differentiate it twice.

Step 1: Differentiate f(x) once with respect to x.
f'(x) = (d/dx)(5/3x^(3)) - (d/dx)(x^(3/2)) + (d/dx)(6)

Differentiating each term separately:
f'(x) = (5/3)(3x^(3-1)) - (3/2)(x^(3/2-1)) + 0

Simplifying:
f'(x) = 5x^2 - (3/2)x^(1/2)

Step 2: Differentiate f'(x) with respect to x to find the second derivative.
f''(x) = (d/dx)(5x^2 - (3/2)x^(1/2))

Again, differentiate each term separately:
f''(x) = (d/dx)(5x^2) - (d/dx)((3/2)x^(1/2))

Differentiating each term:
f''(x) = 10x - (3/4)(1/2)(x^((1/2)-1))

Simplifying:
f''(x) = 10x - (3/4)(1/2)(x^(-1/2))

Further simplification:
f''(x) = 10x - (3/4)(1/2)(1/sqrt(x))

Therefore, the second derivative of f(x) = 5/3x^(3) - x^(3/2) + 6 is f''(x) = 10x - 3/(4sqrt(x)).

To find the second derivative of f(x) = 5/3x^(3) - x^(3/2) + 6, you need to apply the power rule and chain rule of differentiation.

Step 1: Find the first derivative of f(x).
The first derivative of f(x) can be found by differentiating each term separately using the power rule.

The power rule states that if you have a term of the form ax^n, the derivative is given by d/dx(ax^n) = anx^(n-1).

Differentiating the first term (5/3x^3):
Applying the power rule, the derivative is (5/3)(3)x^(3-1) = 5x^2.

Differentiating the second term (-x^(3/2)):
Using the power rule, the derivative is (-3/2)(x^(3/2-1)) = (-3/2)(x^(1/2)) = (-3/2)sqrt(x).

The derivative of the constant term 6 is 0 because the derivative of a constant is always zero.

So, the first derivative (f'(x)) is given by f'(x) = 5x^2 - (3/2)sqrt(x).

Step 2: Find the second derivative of f(x).
To find the second derivative, differentiate the first derivative (f'(x)) obtained from step 1.

Differentiating the first term (5x^2):
Applying the power rule, the derivative is 2(5)x^(2-1) = 10x.

Differentiating the second term (-(3/2)sqrt(x)):
Using the chain rule, the derivative is (d/dx)(-(3/2)sqrt(x)) = -3/4(sqrt(x))^(-1/2) * (1/2x^(-1/2)) = -3/(4sqrt(x)).

Since the derivative of the constant term 0 is 0, it does not appear in the second derivative.

Therefore, the second derivative (f''(x)) is given by f''(x) = 10x - 3/(4sqrt(x)).

Please let me know if I can help you with anything else.