Problem 12 - 5 Steps to A 5: AP Calculus AB 2021

Question

Find the derivative of each of the following functions. $$y=\ln \left(x^{2}+3\right)$$

Step-by-Step Solution

Answer

Question: Find the derivative of the function $$y = \ln \left(x^{2}+3\right)$$. Answer: The derivative of the function is $$y' = \frac{2x}{x^{2}+3}$$.

Step 1: Identify Inner and Outer Functions

We will first identify the inner function (f(x)) and the outer function (g(x)): $$f(x) = x^{2}+3$$ $$g(x) = \ln x$$ The given function can be rewritten as the composition of both functions: $$y = g(f(x)) = \ln \left(x^{2}+3\right)$$

Step 2: Find the Derivative of Inner Function (f(x))

Now, we need to find the derivative of the inner function, f(x): $$f'(x) = \frac{d}{dx}\left( x^{2} + 3 \right)$$ Using the power rule, we find: $$f'(x) = 2x$$

Step 3: Find the Derivative of Outer Function (g(x))

Next, we need to find the derivative of the outer function, g(x), with respect to x: $$g'(x) = \frac{d}{dx}\left( \ln x \right)$$ Using the properties of logarithms, we find: $$g'(x) = \frac{1}{x}$$

Step 4: Apply the Chain Rule

Now that we have the derivatives of both the inner and outer functions, we can use the chain rule to find the derivative of the given function: $$\frac{dy}{dx} = g'(f(x)) \cdot f'(x) = \frac{1}{f(x)} \cdot \left( 2x \right)$$ Using our inner function from step 1, we have: $$\frac{dy}{dx} = \frac{1}{x^{2}+3} \cdot (2x)$$

Step 5: Simplify the Result

Finally, we simplify the derivative: $$\frac{dy}{dx} = \frac{2x}{x^{2}+3}$$ The derivative of the given function is: $$y' = \frac{2x}{x^{2}+3}$$

Key Concepts

Chain Rule

When you're dealing with derivatives, the chain rule can be incredibly useful. It's like a magic formula for finding the derivative of a composition of functions. The chain rule states that if you have two functions, say \( g(x) \) and \( f(x) \), and you want to find the derivative of their composition \( g(f(x)) \), you can do this by:

• First, differentiating the outer function \( g \) with respect to the inner function \( f \), giving you \( g'(f(x)) \).
• Second, multiplying it by the derivative of the inner function \( f(x) \), which is \( f'(x) \).

Putting it all together, you'll have:\[\frac{dy}{dx} = g'(f(x)) \cdot f'(x)\]This allows you to "chain" the derivatives together in a product form, thus making differentiation of composite functions a breeze. Try to visualize each step as you cover more complex compositions.

Power Rule

The power rule is one of the most basic and frequently used rules for differentiation. Whenever you encounter a power of \( x \), such as \( x^n \), you can find its derivative by following a straightforward process:

• Take the exponent \( n \) and bring it down as a coefficient.
• Subtract one from the exponent.

So, the derivative of \( x^n \) is:\[\frac{d}{dx} x^n = n \cdot x^{n-1}\]In our exercise, we used this rule to differentiate the inner function \( x^2 + 3 \), making things quick and efficient. Just remember, for each term in a polynomial, apply the power rule individually!

Logarithmic Differentiation

Logarithmic differentiation is a technique used primarily when dealing with derivatives of logarithmic functions like \( \ln(x) \). It can simplify the differentiation process, especially for complex expressions. Generally, the derivative of the natural logarithm function \( \ln(x) \) is:\[\frac{d}{dx} \ln(x) = \frac{1}{x}\]This means you take the reciprocal of the argument inside the logarithm. In our specific task, this applies to the outer function, \( \ln(x^2+3) \). Remember, when logs are part of a product or a quotient, they simplify the problem significantly through logarithmic properties, also allowing for easier differentiation with the chain rule.

Inner and Outer Functions

In calculus, understanding the distinction between inner and outer functions is crucial, particularly when working with composite functions. A composite function is formed when one function is applied to another, expressed as \( g(f(x)) \). Here, \( f(x) \) is the inner function, and \( g(x) \) is the outer function.

• The inner function is the one "inside" the composition. For \( \ln(x^2 + 3) \), the inner function is \( x^2 + 3 \).
• The outer function is the one "around" the composition. In our case, that's \( \ln(x) \).

Identifying these functions correctly allows you to apply the chain rule effectively. By dissecting the given function into these parts, differentiation can be managed efficiently, helping you solve for derivatives systematically.