Also I would like to know how to set this problem up in finding the answer:
Find the second derivative.
y= sqrt(3x+4)
Thanks, P.S. I tried taking 3x+4 and squaring it to the 1/2. I thought that might've been the same, then I took the derivative of that and got 1/2(3x+4)^(-1/2), then took the 2nd derivative of that and got -1/4(3x+4)^(-1.5). Apparently this isn't right. What am I doing wrong?
y=(3x+4)^(1/2)
y'=(3/2)(3x+4)^(-1/2)
y"=(-9/4)(3x+4)^(-3/2)
i think this is correct. You forgot the chain rule. You also have to multiply by the derivative of 3x+4
THANK U!!
To find the second derivative of y = sqrt(3x + 4), you correctly took the first derivative, but made a mistake in taking the second derivative. Let's go through the correct steps to set up and solve this problem.
Step 1: Take the first derivative using the power rule and chain rule. The power rule states that for a function of the form y = x^n, the derivative is dy/dx = n*x^(n-1). The chain rule states that for a composition of functions, the derivative is the derivative of the outer function times the derivative of the inner function.
For y = sqrt(3x + 4), we can rewrite it as y = (3x + 4)^(1/2). Applying the power rule and chain rule, we have:
y' = (1/2) * (3x + 4)^(-1/2) * d/dx(3x + 4)
Step 2: Find the derivative of the inner function, d/dx(3x + 4). The derivative of 3x + 4 with respect to x is simply 3.
So, y' = (1/2) * (3x + 4)^(-1/2) * 3
Simplifying further, we get:
y' = 3/2 * (3x + 4)^(-1/2)
Step 3: Now, we need to find the second derivative. To do this, we need to take the derivative of y' with respect to x. Applying the power rule and chain rule again, we have:
y'' = d/dx(3/2 * (3x + 4)^(-1/2))
Step 4: Find the derivative of the inner function, d/dx(3x + 4). The derivative of 3x + 4 with respect to x is again 3.
y'' = d/dx(3/2 * (3x + 4)^(-1/2))
= 3/2 * d/dx((3x + 4)^(-1/2))
Step 5: Apply the power rule and chain rule to find the derivative of (3x + 4)^(-1/2).
Using the power rule, the derivative of (3x + 4)^(-1/2) is (-1/2)(3x + 4)^(-3/2). Since we applied the chain rule, we need to multiply this derivative by the derivative of the inner function, which is 3.
y'' = 3/2 * (-1/2)(3x + 4)^(-3/2) * 3
Simplifying further, we get:
y'' = (-9/4)(3x + 4)^(-3/2)
So, the correct second derivative of y = sqrt(3x + 4) is y'' = (-9/4)(3x + 4)^(-3/2).