differentiate using product, chain and quotient rule

ln x / 5x-7

and

e^x / 2x^3 - 1

could you help me out with the first and just point me in the right direction for the first i cant seem to get the steps in the right order and then it gets all muddled

thank you

A!

let u= e^x v=(5x-7)

du/dx=e^x dv/dx= 5
f= uv^-1
f'= u d(v^-1)/dx+ v^-1 du/dx

f'= -uv^-2 dv/dx + v^-1 e^x
so...

f'=-5e^x/(5x-7)^2+ e^x/(5x-7

check that.

Certainly! I can help you with differentiating the given functions using the product, chain, and quotient rules.

First, let's start with the function:

f(x) = ln(x) / (5x - 7)

To differentiate this function, we will use the quotient rule, which states:

If f(x) = g(x) / h(x), then f'(x) = (g'(x)h(x) - g(x)h'(x)) / [h(x)]^2

Now, let's apply the quotient rule to differentiate the function:

Step 1: Identify the numerator (g(x)) and denominator (h(x)).
g(x) = ln(x)
h(x) = 5x - 7

Step 2: Differentiate the numerator and denominator separately.
g'(x) = (1/x) [Using the derivative of ln(x), which is 1/x]
h'(x) = 5 [Derivative of (5x - 7) with respect to x]

Step 3: Apply the quotient rule using the derivatives.
f'(x) = [(g'(x)h(x)) - (g(x)h'(x))] / [h(x)]^2
= [(1/x) * (5x - 7) - ln(x) * 5] / (5x - 7)^2

And that's the derivative of the given function using the quotient rule.

Now, let's move on to the function:

g(x) = e^x / (2x^3 - 1)

To differentiate this function, we will again use the quotient rule.

However, before applying the quotient rule, we need to rewrite the function using the chain rule.

The function can be written as:

g(x) = e^x * (2x^3 - 1)^(-1)

Now, the function is in the form of a product. To differentiate this using the product rule:

Step 1: Identify the two factors.
g(x) = e^x
h(x) = (2x^3 - 1)^(-1)

Step 2: Differentiate each factor separately.
g'(x) = e^x [Derivative of e^x with respect to x]
h'(x) = -3(2x^3 - 1)^(-2) * (6x^2) [Chain rule: derivative of (2x^3 - 1) to the power of -1]

Step 3: Apply the product rule to the derivative.
g'(x)h(x) + g(x)h'(x)
= e^x * (2x^3 - 1)^(-1) + e^x * [-3(2x^3 - 1)^(-2) * (6x^2)]
= e^x * (2x^3 - 1)^(-1) - 18x^2e^x / (2x^3 - 1)^2

And that's the derivative of the given function using the product rule.