Chain Rule Demystified Solving Y=e^x(3x-1)^4 With Calculus
In the realm of calculus, the chain rule stands as a cornerstone technique for differentiating composite functions. It empowers us to unravel the intricate layers of functions nested within one another, allowing us to determine their derivatives with precision. This comprehensive guide delves into the application of the chain rule to solve the derivative of the function y=ex(3x-1)4, providing a step-by-step walkthrough and elucidating the underlying principles.
Understanding the Chain Rule: A Foundation for Differentiation
At its core, the chain rule provides a systematic approach to differentiating composite functions, which are functions formed by nesting one function inside another. Mathematically, it states that the derivative of a composite function f(g(x)) is given by:
d/dx [f(g(x))] = f'(g(x)) * g'(x)
In simpler terms, the chain rule instructs us to differentiate the outer function f with respect to the inner function g(x), and then multiply the result by the derivative of the inner function g'(x). This iterative process allows us to peel away the layers of the composite function, differentiating each layer in turn.
To truly grasp the chain rule, let's consider an analogy. Imagine a set of Russian nesting dolls, where each doll is encased within a larger one. To access the smallest doll at the center, you must sequentially remove each outer doll. Similarly, when differentiating a composite function, the chain rule guides us to differentiate each layer of the function, starting from the outermost layer and working our way inwards.
The power of the chain rule lies in its ability to handle complex functions that would otherwise be difficult or impossible to differentiate directly. By breaking down these functions into simpler components, the chain rule transforms the differentiation process into a manageable series of steps.
Deconstructing y=ex(3x-1)4: Identifying the Composite Functions
Before we embark on the differentiation process, let's dissect the function y=ex(3x-1)4 to identify the composite functions at play. This function is a product of two distinct components: the exponential function e^x and the power function (3x-1)^4. The power function itself is a composite function, where the inner function is (3x-1) and the outer function is raising to the power of 4.
This identification of composite functions is crucial, as it dictates the application of the chain rule. We must treat each composite function as a separate entity, differentiating it using the chain rule before combining the results.
To further clarify, let's break down the function y=ex(3x-1)4 into its constituent parts:
- Outer function: The product of e^x and (3x-1)^4
- Inner function 1: e^x (a simple exponential function)
- Inner function 2: (3x-1)^4 (a composite power function)
- Innermost function: 3x-1 (a linear function within the power function)
With this decomposition in mind, we can now confidently apply the chain rule to differentiate each layer of the function.
Applying the Chain Rule: A Step-by-Step Differentiation
Now, let's embark on the journey of differentiating y=ex(3x-1)4 using the chain rule. We'll proceed step-by-step, meticulously applying the rule to each composite function.
Step 1: Differentiate the Outer Function (Product Rule)
Since our outer function is a product of two components, e^x and (3x-1)^4, we must first employ the product rule. The product rule states that the derivative of the product of two functions, u(x) and v(x), is given by:
d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
In our case, let u(x) = e^x and v(x) = (3x-1)^4. Applying the product rule, we get:
d/dx [ex(3x-1)4] = (d/dx ex)(3x-1)4 + e^x (d/dx (3x-1)^4)
Step 2: Differentiate e^x (Simple Exponential Function)
The derivative of e^x is simply e^x. This is a fundamental rule of differentiation and requires no further application of the chain rule.
d/dx e^x = e^x
Step 3: Differentiate (3x-1)^4 (Composite Power Function)
Now, we encounter a composite function, (3x-1)^4, where the chain rule truly shines. We'll apply the chain rule to differentiate this function.
Let's break down the chain rule application:
- Outer function: Raising to the power of 4 (let's represent it as f(u) = u^4)
- Inner function: 3x-1 (let's represent it as g(x) = 3x-1)
Applying the chain rule, we get:
d/dx (3x-1)^4 = d/dx [f(g(x))] = f'(g(x)) * g'(x)
First, we differentiate the outer function f(u) = u^4 with respect to u:
f'(u) = 4u^3
Next, we substitute the inner function g(x) = 3x-1 for u:
f'(g(x)) = 4(3x-1)^3
Now, we differentiate the inner function g(x) = 3x-1 with respect to x:
g'(x) = 3
Finally, we multiply f'(g(x)) by g'(x) to obtain the derivative of (3x-1)^4:
d/dx (3x-1)^4 = 4(3x-1)^3 * 3 = 12(3x-1)^3
Step 4: Combine the Results
Now that we've differentiated each component, we can combine the results to obtain the derivative of the original function y=ex(3x-1)4.
Recall from Step 1 that:
d/dx [ex(3x-1)4] = (d/dx ex)(3x-1)4 + e^x (d/dx (3x-1)^4)
Substituting the derivatives we calculated in Steps 2 and 3, we get:
d/dx [ex(3x-1)4] = ex(3x-1)4 + e^x [12(3x-1)^3]
Step 5: Simplify the Expression (Optional)
We can further simplify the expression by factoring out common terms:
d/dx [ex(3x-1)4] = ex(3x-1)3 [(3x-1) + 12]
Simplifying the expression within the brackets, we get:
d/dx [ex(3x-1)4] = ex(3x-1)3 (3x + 11)
Therefore, the derivative of y=ex(3x-1)4 is:
dy/dx = ex(3x-1)3 (3x + 11)
Conclusion: Mastering the Chain Rule for Complex Differentiation
In this comprehensive guide, we've meticulously dissected the application of the chain rule to solve the derivative of the function y=ex(3x-1)4. By breaking down the function into its composite components and systematically applying the chain rule, we've successfully navigated the complexities of differentiation.
The chain rule stands as a powerful tool in the calculus toolkit, enabling us to differentiate a wide range of composite functions. By mastering the chain rule, you unlock the ability to tackle intricate differentiation problems with confidence and precision. This step-by-step walkthrough serves as a testament to the chain rule's effectiveness and empowers you to apply it to diverse mathematical scenarios. Remember, practice is key to mastering any mathematical technique, so embrace the challenge and continue honing your skills in the realm of calculus.
How to use the chain rule to find the derivative of y = ex(3x-1)4?
Chain Rule Demystified Solving y=ex(3x-1)4 with Calculus
