What would an example of a logarithmic function that contains a radical within it and a quotient within it where you would need to use chain rule and quotient rule to take derivative look like? How would taking the derivative of the function in its original form look different from use log properties to simplify the function first and then take derivative?

Question

What would an example of a logarithmic function that contains a radical within it and a quotient within it where you would need to use chain rule and quotient rule to take derivative look like? How would taking the derivative of the function in its original form look different from use log properties to simplify the function first and then take derivative?

Answer 1

did you mean something like

y = √( (x^2 + 5x)/(5x) )

original form derivative:
dy/dx= (1/2)((x^2+5)/(5x))^(-1/2) ( 5x(2x) - (x^2+5)(5))/(25x^2)
etc , but still has to be simplified

using logs first:
take ln of both sides:
ln y = ln((x^+5)/(5x))^(1/2)
= (1/2)(ln (x^2+5) - ln (5x) )

(dy/dx)/y = (1/2)( 2x/(x^2+5) - 5/(5x) )
= x/(x^2 + 5) - x/2

dy/dx = y( x/(x^2 + 5) - x/2 )
or
= (√( (x^2 + 5x)/(5x) )( x/(x^2 + 5) - x/2 )

this may not look any simpler, but it was easier to get to.

Answer 2

An example of a logarithmic function that contains a radical within it and a quotient within it, where you would need to use the chain rule and quotient rule to take the derivative, could be:

f(x) = ln(sqrt(x)) / x

To take the derivative of this function in its original form, you would need to apply the chain rule and quotient rule simultaneously.

Applying the chain rule, we begin by differentiating the outermost function. In this case, the outermost function is the natural logarithm (ln). The derivative of ln(u) with respect to x is du/dx divided by u, where u is the inner function. Therefore, the derivative of ln(sqrt(x)) is 1 / (2 * sqrt(x)).

Next, we apply the quotient rule. The quotient rule states that if you have a function f(x) = g(x) / h(x), then its derivative is (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2.

For our given function f(x) = ln(sqrt(x)) / x, let's evaluate each part of the quotient rule:

g(x) = ln(sqrt(x))
g'(x) = 1 / (2 * sqrt(x))

h(x) = x
h'(x) = 1

Now, using the quotient rule, we have:

f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
= ((1 / (2 * sqrt(x))) * x - ln(sqrt(x)) * 1) / x^2
= (x / (2 * sqrt(x)) - ln(sqrt(x))) / x^2
= (x - 2 * ln(sqrt(x))) / (2 * x^2 * sqrt(x))

Now, let's see how the process would look different if we simplify the function using log properties first, and then take the derivative.

Original function: f(x) = ln(sqrt(x)) / x

First, we can simplify the function using log properties:

f(x) = ln(sqrt(x)) / x
= (1/2) * ln(x) / x

Now, we can take the derivative of this simplified expression. This will be easier compared to the original form since we no longer have a radical within the natural logarithm.

Applying the quotient rule, we have:

f'(x) = ((1/2) * (1/x) * x - ln(x) * 1) / x^2
= (1/2 - ln(x)) / x^2

Therefore, the derivative of the simplified function is given by (1/2 - ln(x)) / x^2.

In summary, taking the derivative of the function in its original form requires using both the chain rule and the quotient rule. On the other hand, simplifying the function using log properties first allows us to take the derivative more straightforwardly, only using the quotient rule.