Find (d^(2))/(dx^(2))[ln(2x+4)].

Question

Find  (d^(2))/(dx^(2))[ln(2x+4)].

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Find First Derivative: We need to find the second derivative of the function f(x) = ln(2x+4) with respect to x. The first step is to find the first derivative f'(x) using the chain rule. The chain rule states that if you have a composite function f(g(x)), then the derivative f'(x) is f'(g(x)) * g'(x). In our case, f(g(x)) = ln(2x+4), so g(x) = 2x+4 and f(g(x)) = ln(g(x)). Now we differentiate f(g(x)) with respect to g(x) and then multiply by the derivative of g(x) with respect to x. f'(g(x)) = 1/g(x) since the derivative of ln(g(x)) with respect to g(x) is 1/g(x). g'(x) = 2 since the derivative of 2x+4 with respect to x is 2. So, f'(x) = f'(g(x)) * g'(x) = (1/(2x+4)) * 2.Simplify First Derivative: Now we simplify the first derivative f'(x). f'(x) = (1/(2x+4)) * 2 = 2/(2x+4). We can simplify this further by dividing the numerator and the denominator by 2. f'(x) = 1/(x+2).Find Second Derivative: Next, we find the second derivative f''(x) by differentiating f'(x) with respect to x. The derivative of 1/(x+2) with respect to x is found using the quotient rule or by recognizing it as a power of x. We can rewrite 1/(x+2) as (x+2)^(-1) and then differentiate. Using the power rule, the derivative of (x+2)^(-1) with respect to x is -1 * (x+2)^(-2) * 1, since the derivative of (x+2) with respect to x is 1. So, f''(x) = -1 * (x+2)^(-2).Simplify Second Derivative: We can leave the second derivative in its current form or simplify it further. f''(x) = -1 * (x+2)^(-2) can be rewritten as -1/(x+2)^2. This is the simplified form of the second derivative.

Find $\frac{d^{2}}{d x^{2}}[\ln (2 x+4)]$ .

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Find d2dx2[ln⁡(2x+4)] \frac{d^{2}}{d x^{2}}[\ln (2 x+4)] dx2d2​[ln(2x+4)].

Find $\frac{d^{2}}{d x^{2}}[\ln (2 x+4)]$ .