Find the second derivative.
y = sqrt(3x+4)
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I am an AI bot named "Explain Bot." I am here to assist with providing explanations and helping you find the answer to your question. For the problem you mentioned, finding the second derivative of the function y = √(3x + 4), we can follow the steps below:
Step 1: Start with the given function y = √(3x + 4).
Step 2: Use the power rule for differentiation, which states that if y = x^n, then dy/dx = n*x^(n-1). In this case, since we have a square root, we can rewrite the function as y = (3x + 4)^(1/2).
Step 3: Apply the chain rule, which states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x), where f'(x) represents the derivative of f(x) and g'(x) represents the derivative of g(x).
In this case, f(u) = u^(1/2) and g(x) = 3x + 4.
Step 4: Find the derivative of f(u) using the power rule: f'(u) = (1/2) * u^(-1/2).
Step 5: Find the derivative of g(x) using the power rule: g'(x) = 3.
Step 6: Apply the chain rule: dy/dx = f'(g(x)) * g'(x).
Substituting the expressions from steps 4 and 5, we have dy/dx = (1/2) * (3x + 4)^(-1/2) * 3.
Step 7: Simplify the expression: dy/dx = (3/2) * (3x + 4)^(-1/2).
Step 8: Find the second derivative by differentiating dy/dx with respect to x.
Using the power rule, we differentiate the expression in step 7, which gives us d^2y/dx^2 = d/dx[(3/2) * (3x + 4)^(-1/2)].
Step 9: Simplify the expression: d^2y/dx^2 = (3/2) * (-1/2) * (3x + 4)^(-3/2) * 3.
Combining the coefficients, we have d^2y/dx^2 = -9/4 * (3x + 4)^(-3/2).
So, the second derivative of y = √(3x + 4) is d^2y/dx^2 = -9/4 * (3x + 4)^(-3/2).