Question

Calculate the derivative of the following functions. $$y=\left(\frac{3 x}{4 x+2}\right)^{5}$$

Short answer

Question: Find the derivative of the function $$y = \Big(\frac{3x}{4x+2}\Big)^5$$ with respect to x. Answer: The derivative of the given function is $$\frac{dy}{dx} = 5\Big(\frac{3x}{4x+2}\Big)^4 \cdot \frac{6}{(4x+2)^2}$$.

Step-by-step solution

Identify the outer and inner functions

We have a composition of functions with the form: $$y = (\frac{3x}{4x+2})^5$$ In this case, we can identify that: - Outer function: \(u^5\), where \(u\) is the inner function. - Inner function: \(u = \frac{3x}{4x+2}\) Now let's differentiate the outer function with respect to \(u\) and the inner function with respect to \(x\).

Compute the derivative of the outer function

The outer function is \(u^5\). We'll differentiate this with respect to \(u\), using the power rule: $$\frac{d}{du}(u^5) = 5u^4$$

Compute the derivative of the inner function

The inner function is \(u = \frac{3x}{4x+2}\). We'll differentiate this with respect to \(x\), using the quotient rule: $$\frac{d}{dx}\Big(\frac{3x}{4x+2}\Big) = \frac{(3)(4x+2)-(3x)(4)}{(4x+2)^2} = \frac{6}{(4x+2)^2}$$

Apply the chain rule

Now we'll apply the chain rule, which states that: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$ We already have both \(\frac{dy}{du} = 5u^4\) and \(\frac{du}{dx} = \frac{6}{(4x+2)^2}\), so we can plug them in: $$\frac{dy}{dx} = 5u^4 \cdot \frac{6}{(4x+2)^2}$$

Substitute the inner function back into the result

The final step is to replace \(u\) with the expression from the inner function and simplify the result: $$\frac{dy}{dx} = 5\Big(\frac{3x}{4x+2}\Big)^4 \cdot \frac{6}{(4x+2)^2}$$ Thus, the derivative of the given function is: $$\frac{dy}{dx} = 5\Big(\frac{3x}{4x+2}\Big)^4 \cdot \frac{6}{(4x+2)^2}$$

Key concepts

Chain Rule

The chain rule is an essential technique in calculus for differentiating compositions of functions. Whenever you have a function nested within another, like in our case with \(y = \left(\frac{3x}{4x+2}\right)^5\), the chain rule helps find the derivative efficiently.

Here's how it works:

• Identify the outer function. In our exercise, it's \(u^5\), where \(u = \frac{3x}{4x+2}\).
• Differentiate the outer function with respect to the inner function. This results in \(\frac{dy}{du} = 5u^4\).
• Next, differentiate the inner function \(u\) with respect to \(x\). For the given function, it is \(\frac{6}{(4x+2)^2}\).
• Finally, multiply these derivatives together to apply the chain rule: \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\).

This allows us to find the derivative of complex, nested functions with ease.

Power Rule

The power rule is a straightforward and powerful tool for finding derivatives of functions that raise a variable to a power. It states that if you have a function of the form \(f(x) = x^n\), its derivative is given by: \(f'(x) = nx^{n-1}\).

In our problem, the power rule is used when differentiating the outer function \((u^5)\). This calculation gives us:

• Start with \(u^5\).
• Differentiating using the power rule yields \(5u^4\).

By applying the power rule, you can quickly handle outer functions raised to any power, simplifying the overall differentiation process when combined with other rules.

Quotient Rule

The quotient rule comes into play when you're dealing with a function that is the ratio of two differentiable functions. Suppose you have \(f(x) = \frac{g(x)}{h(x)}\). Then the derivative, \(f'(x)\), is given by:\[\frac{d}{dx} \left( \frac{g(x)}{h(x)} \right) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}.\]In this exercise, the quotient rule is used to find the derivative of the inner function \(\frac{3x}{4x+2}\), which involves these steps:

• Let \(g(x) = 3x\) and \(h(x) = 4x+2\).
• Differentiate \(g(x)\) and \(h(x)\): \(g'(x) = 3\) and \(h'(x) = 4\).
• Plug these into the quotient rule formula: \(\frac{d}{dx} \left( \frac{3x}{4x+2} \right) = \frac{3(4x+2) - 3x(4)}{(4x+2)^2} = \frac{6}{(4x+2)^2}\).

Using the quotient rule allows us to tackle functions expressed as fractions, maintaining their integrity while finding their rates of change.