why do we use chain rule for deriving ln(x^2+5x)
so wrong!
Let u = x^2 + 5x
Then y = ln(u)
so
y' = 1/u u' = (2x+5)/(x^2+5x)
Apologies for the mistake in the previous response. Thank you for providing the correct derivation using the chain rule.
To find the derivative of ln(x^2 + 5x), we can let u = x^2 + 5x. Then, y = ln(u) which gives us y = ln(x^2 + 5x).
Applying the chain rule to find the derivative:
dy/dx = dy/du * du/dx
Since y = ln(u), dy/du = 1/u and u = x^2 + 5x, so du/dx = 2x + 5.
Therefore,
dy/dx = 1/u * (2x + 5)
= (2x + 5) / (x^2 + 5x)
This is the correct derivative of ln(x^2 + 5x) using the chain rule. Thank you for pointing out the error.
We use the chain rule for deriving ln(x^2 + 5x) because it is a composite function. The function ln(x) is the outer function and x^2 + 5x is the inner function. When we take the derivative of ln(x^2 + 5x) with respect to x, we need to use the chain rule to differentiate the composite function.
In other words, the chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In this case, the outer function is ln(x) and the inner function is x^2 + 5x.
So when we apply the chain rule to ln(x^2 + 5x), we differentiate ln(x) with respect to x (which is 1/x) and then multiply it by the derivative of the inner function (which is 2x + 5). This gives us the final derivative of ln(x^2 + 5x) as:
(1/x) * (2x + 5) = 2 + 5/x