Question

Find \(f^{\prime}(x)\) for each function. $$ f(x)=\log _{7}\left(6 x^{4}+3\right)^{5} $$

Short answer

The derivative is \( f^{\prime}(x) = \frac{120x^3(6x^4 + 3)^4}{\ln(7)} \).

Step-by-step solution

Identify the Components

First, we identify the components of the function. The function given is a composite function involving a logarithmic function and a power function. The outer function is the logarithm, and the inner function is \(g(x) = (6x^4 + 3)^5\).

Apply the Chain Rule

To differentiate the composite function \log_7((6x^4 + 3)^5)\, we use the chain rule. The derivative \(f^{\prime}(x)\) will be the derivative of the outer function evaluated at the inner function times the derivative of the inner function: \[f^{\prime}(x) = \frac{d}{dx} \left[\log_7(u)\right] \cdot \frac{d}{dx}[u]\] where \u = (6x^4 + 3)^5\.

Differentiate the Logarithmic Function

The derivative of \log_7(u)\ with respect to \u\ is \[\frac{1}{u \ln(7)}\].

Differentiate the Power Function

The inner function \(u = (6x^4 + 3)^5\) is itself a power function. We differentiate it using the chain rule:\[\frac{d}{dx}(6x^4 + 3)^5 = 5(6x^4 + 3)^4 \times \frac{d}{dx}(6x^4 + 3).\]

Differentiate Inside the Power Function

Now differentiate \(6x^4 + 3\) with respect to \x\: \[\frac{d}{dx}(6x^4 + 3) = 24x^3.\]

Combine All Derivatives

Substitute the derivatives found in previous steps into the chain rule formula: \[f^{\prime}(x) = \frac{1}{(6x^4 + 3)^5 \ln(7)} \times 5(6x^4 + 3)^4 \times 24x^3.\]Simplify this expression: \[f^{\prime}(x) = \frac{5 \cdot 24x^3 \cdot (6x^4 + 3)^4}{\ln(7)}.\]

Final Simplification

Finally, calculate the constant factor \(5 \cdot 24\) to simplify the expression: \[f^{\prime}(x) = \frac{120x^3(6x^4 + 3)^4}{\ln(7)}.\]

Key concepts

Chain Rule

The Chain Rule is a fundamental tool in calculus for finding the derivative of composite functions. A composite function is essentially a function within another function. In this exercise, the function is given as \( \log_{7}((6x^4 + 3)^5) \). Here, the chain rule becomes essential because we're dealing with not just a basic function but a combination of a logarithmic and a power function.
To apply the Chain Rule, we first identify both the outer and the inner functions:

• The outer function is the logarithmic function \( \log_{7}(u) \).
• The inner function, \( u \), is \( (6x^4 + 3)^5 \).

The Chain Rule states that to find the derivative of a composite function, you must multiply the derivative of the outer function by the derivative of the inner function. It's like peeling an onion: you take the outer layer first, then work your way inward. By applying this rule step-by-step, we ensure each element of the function is accounted for in its differentiation.
Using the chain rule helps in breaking down complex problems into manageable steps, making calculus more accessible.

Derivative of Logarithmic Function

When differentiating logarithmic functions, like \( \log_{7}(u) \), it's important to understand the general rule for logarithmic differentiation. The derivative of a logarithm with a base other than \( e \) is measured using the formula:

• \( \frac{d}{dx} [\log_{a}(u)] = \frac{1}{u \ln(a)} \cdot \frac{du}{dx} \)

In our exercise, the base of the logarithm is 7, so the derivative of \( \log_7(u) \) is \( \frac{1}{u \ln(7)} \). It’s important to remember this additional \( \ln(7) \) term in the denominator, which comes from the change of base in logarithms.
The key takeaway here is that while handling derivatives involving logarithmic functions, different bases require adjusting your derivative with the natural logarithm of the base \( a \). This adjustment ensures the derivative's accuracy according to the basis used, offering a consistent approach to solving differentiation problems involving logs.

Power Function Differentiation

Power functions often appear in calculus, and their differentiation follows a predictable pattern known as the power rule. Here, the inner function is \( (6x^4 + 3)^5 \), and it’s a perfect example of a power function.
The power rule states that if we have a function \( u(x) = x^n \), then its derivative is \( nu^{n-1} \cdot \frac{du}{dx} \). However, since our inner function is more complex, we combine the power rule with the chain rule for effective differentiation.
Here are the steps involved:

• First, apply the power rule to \( (6x^4 + 3)^5 \) by bringing down the 5, resulting in \( 5(6x^4 + 3)^4 \).
• Next, differentiate \( 6x^4 + 3 \), yielding \( 24x^3 \).

Finally, the derivative of the inner function can be expressed as \( 5(6x^4 + 3)^4 \times 24x^3 \). This method helps us manage power functions within larger composite expressions effectively, ensuring we accurately calculate their derivatives across different scenarios.