Derivative Of Log Function: Step-by-Step Calculation

Hey guys! Today, we're diving into the exciting world of calculus to tackle a common problem: finding the derivative of a logarithmic function. Specifically, we're going to calculate the derivative of the function y = 7 * log₄(x⁴ - 5). Buckle up, because we're about to break this down step-by-step so it's super easy to understand. Trust me; by the end of this, you'll be a pro at differentiating logarithmic functions!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what we're trying to achieve. The problem asks us to find the derivative of the function y = 7 * log₄(x⁴ - 5) with respect to x. In simpler terms, we want to know how the value of y changes as x changes. This is a fundamental concept in calculus, and it has numerous applications in physics, engineering, economics, and many other fields. The derivative will give us a formula for the instantaneous rate of change of the function at any given point.

The given function involves a logarithmic function with base 4, multiplied by a constant (7), and the argument of the logarithm is a polynomial function (x⁴ - 5). To find the derivative, we'll need to apply the chain rule, which is a rule for differentiating composite functions. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. We'll also need to know the derivative of a logarithm function with a general base. Let's refresh some key concepts before diving into the actual calculation.

Logarithmic Functions: A logarithmic function is the inverse of an exponential function. The function logₐ(x) gives the exponent to which the base 'a' must be raised to produce the value 'x'.

Derivatives: The derivative of a function measures the instantaneous rate of change of the function. It provides the slope of the tangent line to the function's graph at any given point.

Chain Rule: The chain rule is used to differentiate composite functions. If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Derivative of Logarithmic Functions: The derivative of logₐ(x) is 1 / (x * ln(a)), where ln(a) is the natural logarithm of a.

With these concepts in mind, we are well-equipped to solve the problem at hand. Let's proceed with the step-by-step solution.

Step-by-Step Solution

1. Identify the Components

First, let's break down the function y = 7 * log₄(x⁴ - 5) into its components. We have:

• A constant multiplier: 7
• A logarithmic function: log₄(u), where u = x⁴ - 5
• An inner function: u = x⁴ - 5

2. Apply the Constant Multiple Rule

The constant multiple rule states that if you have a constant multiplied by a function, the derivative is simply the constant multiplied by the derivative of the function. In our case:

d/dx [7 * log₄(x⁴ - 5)] = 7 * d/dx [log₄(x⁴ - 5)]

3. Apply the Chain Rule

Now, we need to find the derivative of log₄(x⁴ - 5). Here, we'll use the chain rule. Recall that the chain rule states d/dx [f(g(x))] = f'(g(x)) * g'(x). In our case:

• Outer function: f(u) = log₄(u)
• Inner function: g(x) = x⁴ - 5

So we need to find f'(u) and g'(x).

4. Find the Derivative of the Outer Function

The derivative of the logarithmic function logₐ(u) with respect to u is given by:

d/du [logₐ(u)] = 1 / (u * ln(a))

In our case, a = 4, so:

d/du [log₄(u)] = 1 / (u * ln(4))

Replacing u with (x⁴ - 5), we get:

f'(g(x)) = 1 / ((x⁴ - 5) * ln(4))

5. Find the Derivative of the Inner Function

Now, we need to find the derivative of the inner function g(x) = x⁴ - 5 with respect to x:

g'(x) = d/dx [x⁴ - 5]

Using the power rule, which states d/dx [xⁿ] = n * xⁿ⁻¹, we get:

g'(x) = 4x³

6. Apply the Chain Rule (Putting it Together)

Now we multiply the derivative of the outer function evaluated at the inner function by the derivative of the inner function:

d/dx [log₄(x⁴ - 5)] = f'(g(x)) * g'(x) = [1 / ((x⁴ - 5) * ln(4))] * 4x³

So,

d/dx [log₄(x⁴ - 5)] = 4x³ / ((x⁴ - 5) * ln(4))

7. Apply the Constant Multiple Rule Again

Remember that we have the constant multiplier 7, so we multiply the result by 7:

d/dx [7 * log₄(x⁴ - 5)] = 7 * [4x³ / ((x⁴ - 5) * ln(4))]

Which simplifies to:

d/dx [7 * log₄(x⁴ - 5)] = 28x³ / ((x⁴ - 5) * ln(4))

Final Answer

So, the derivative of the function y = 7 * log₄(x⁴ - 5) is:

dy/dx = 28x³ / ((x⁴ - 5) * ln(4))

Common Mistakes to Avoid

When dealing with derivatives, especially those involving logarithmic functions and the chain rule, it's easy to make mistakes. Here are a few common pitfalls to watch out for:

• Forgetting the Chain Rule: The chain rule is essential when differentiating composite functions. Make sure you apply it correctly by multiplying the derivative of the outer function by the derivative of the inner function.
• Incorrect Derivative of Logarithmic Functions: The derivative of logₐ(x) is 1 / (x * ln(a)). It's easy to forget the natural logarithm of the base or to mix it up with other formulas.
• Algebraic Errors: Be careful when simplifying expressions. A small algebraic error can lead to an incorrect final answer.
• Not Applying the Constant Multiple Rule: If you have a constant multiplied by a function, remember to multiply the derivative of the function by the constant.

Practice Problems

To solidify your understanding, here are a few practice problems:

1. Find the derivative of y = 5 * log₂(x³ + 2).
2. Calculate dy/dx for y = 3 * log₅(2x² - 1).
3. Differentiate y = 8 * log₃(x⁵ + 3).

Work through these problems, and you'll become even more confident in your ability to differentiate logarithmic functions. These exercises will help reinforce the steps and techniques we've covered, ensuring you can apply them effectively in various scenarios.

Conclusion

And there you have it! We've successfully calculated the derivative of the function y = 7 * log₄(x⁴ - 5). By breaking down the problem into smaller, manageable steps, applying the chain rule, and being mindful of common mistakes, we were able to arrive at the correct answer:

dy/dx = 28x³ / ((x⁴ - 5) * ln(4))

Calculus might seem daunting at first, but with practice and a clear understanding of the rules and concepts, you can conquer any problem. Keep practicing, and you'll become a calculus wizard in no time! Remember, the key is to understand the underlying principles, practice regularly, and don't be afraid to ask for help when you need it. Happy calculating, and remember to stay curious! High-quality content is key to mastering any skill, so keep exploring and expanding your knowledge.