Question

Find the derivative g(x)=ln(2x^2+1)

Answer 1

To find the derivative of g(x) = ln(2x^2 + 1), use the chain rule: g'(x) = 1 / (2x^2 + 1) * (4x).

Step-by-step explanation:

To find the derivative of the function g(x) = ln(2x^2 + 1), we can use the chain rule. The chain rule states that if we have a composite function, such as ln(f(x)), the derivative is given by the derivative of the outer function (ln) multiplied by the derivative of the inner function (f'(x)). In this case, the derivative of ln(2x^2 + 1) is:

1. Take the derivative of the outer function: g'(x) = 1 / (2x^2 + 1)
2. Multiply this by the derivative of the inner function, which is the derivative of (2x^2 + 1): g'(x) = 1 / (2x^2 + 1) * (4x)
3. Simplify the expression if needed.

Answer 2

4x / (2x^2+1)

Step-by-step explanation:

g(x)=ln(2x^2+1)

Using u substitution

u= 2x^2 +1

du = 4x dx

g'(x) = d/du ( g(u) du)

We know that the derivative of ln(u) = 1/u since this is always positive

= 1/ u* du

Substituting x back into the equation

g'(x) = 1/(2x^2+1) * 4x

= 4x / (2x^2+1)