Derivative Of Y=(3x^-3 + 2x)^3 With Respect To X

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In this article, we will delve into the process of finding the derivative of the function $y = (3x^{-3} + 2x)^3$ with respect to x. This problem falls under the domain of calculus, specifically the application of the chain rule. The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. A composite function is essentially a function within another function, and in our case, we have the function $3x^{-3} + 2x$ nested inside the power function $(...)^3$. Understanding and applying the chain rule correctly is crucial for solving this type of problem. Let's break down the steps involved in finding the derivative and explore the underlying concepts in detail.

Understanding the Chain Rule

The chain rule is a powerful tool in calculus used to differentiate composite functions. A composite function is a function that is composed of another function. Mathematically, if we have a function $y = f(g(x))$, then the derivative of y with respect to x, denoted as $\frac{dy}{dx}$, is given by:

dydx=dfdgâ‹…dgdx\frac{dy}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx}

In simpler terms, the chain rule states that the derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function. To effectively use the chain rule, it's essential to correctly identify the outer and inner functions within the composite function. The outer function is the function that acts on the entire expression, while the inner function is the function that is nested inside the outer function. Recognizing this structure is the first step in successfully applying the chain rule. For instance, in our problem $y = (3x^{-3} + 2x)^3$, the outer function is the power function $(...)^3$, and the inner function is the expression $3x^{-3} + 2x$. The chain rule essentially allows us to "peel away" the layers of the composite function, differentiating each layer separately and then multiplying the results together. This systematic approach ensures that we account for the contribution of each function in the composition to the overall derivative.

Applying the Chain Rule to Our Function

Now, let's apply the chain rule to find the derivative of $y = (3x^{-3} + 2x)^3$. First, we identify the outer and inner functions. As we mentioned earlier, the outer function is $u^3$, where $u$ represents the inner function. The inner function is $u = 3x^{-3} + 2x$. Now, we need to find the derivatives of both the outer and inner functions separately.

Derivative of the Outer Function

The derivative of the outer function, $u^3$, with respect to $u$ is found using the power rule. The power rule states that if we have a function of the form $f(x) = x^n$, where n is a constant, then its derivative is $f'(x) = nx^{n-1}$. Applying the power rule to our outer function, we get:

d(u3)du=3u2\frac{d(u^3)}{du} = 3u^2

This is a straightforward application of the power rule, where we multiply the exponent (3) by the coefficient (1, implied) and then reduce the exponent by 1 (3 - 1 = 2). This result represents the rate of change of the outer function with respect to the inner function $u$. This is a crucial component in applying the chain rule, as we will need to multiply this derivative by the derivative of the inner function to obtain the overall derivative of the composite function.

Derivative of the Inner Function

Next, we need to find the derivative of the inner function, $u = 3x^{-3} + 2x$, with respect to $x$. This requires us to differentiate each term in the expression separately. We will again use the power rule for this. Let's break it down:

  • Derivative of $3x^-3}$ Applying the power rule, we multiply the coefficient (3) by the exponent (-3) and then reduce the exponent by 1 (-3 - 1 = -4). This gives us $3 \cdot (-3)x^{-4 = -9x^{-4}$.
  • Derivative of $2x$: This is a simpler application of the power rule. We multiply the coefficient (2) by the exponent (1, implied) and then reduce the exponent by 1 (1 - 1 = 0). This gives us $2 \cdot 1x^0 = 2$, since any non-zero number raised to the power of 0 is 1.

Combining these results, we get the derivative of the inner function:

d(3x−3+2x)dx=−9x−4+2\frac{d(3x^{-3} + 2x)}{dx} = -9x^{-4} + 2

This derivative represents the rate of change of the inner function with respect to $x$. It captures how the inner function changes as $x$ changes. Now that we have the derivatives of both the outer and inner functions, we can apply the chain rule to find the derivative of the original composite function.

Applying the Chain Rule Formula

Now that we have the derivatives of both the outer and inner functions, we can apply the chain rule formula:

dydx=dfduâ‹…dudx\frac{dy}{dx} = \frac{df}{du} \cdot \frac{du}{dx}

Substituting the derivatives we found earlier:

dydx=3u2⋅(−9x−4+2)\frac{dy}{dx} = 3u^2 \cdot (-9x^{-4} + 2)

Remember that $u = 3x^{-3} + 2x$. We need to substitute this back into the expression to get the derivative in terms of $x$ only:

dydx=3(3x−3+2x)2⋅(−9x−4+2)\frac{dy}{dx} = 3(3x^{-3} + 2x)^2 \cdot (-9x^{-4} + 2)

This is the derivative of the function $y = (3x^{-3} + 2x)^3$ with respect to $x$. We have successfully applied the chain rule by differentiating the outer function, differentiating the inner function, and then multiplying the results together. The final step is often to simplify the expression, if possible.

Simplifying the Derivative (Optional)

While the expression we obtained in the previous step is the correct derivative, it can often be simplified for clarity or further use. Let's simplify the derivative we found:

dydx=3(3x−3+2x)2⋅(−9x−4+2)\frac{dy}{dx} = 3(3x^{-3} + 2x)^2 \cdot (-9x^{-4} + 2)

We can expand the expression, but it will result in a more complex polynomial. It's often beneficial to leave it in this factored form unless further operations (like finding critical points) are required. However, let's distribute the $3$ inside the first term to get a slightly simplified form:

dydx=3(3x−3+2x)2(2−9x−4)\frac{dy}{dx} = 3(3x^{-3} + 2x)^2 (2 - 9x^{-4})

This form is slightly more compact and may be preferred in some contexts. The key is to understand the context in which you are using the derivative. If you need to find critical points, setting the derivative to zero might be easier in the factored form. If you need to perform further algebraic manipulations, a more expanded form might be beneficial. However, for most purposes, the factored form is sufficient and often preferred.

Conclusion

In this article, we successfully found the derivative of the function $y = (3x^{-3} + 2x)^3$ with respect to $x$ using the chain rule. We first understood the chain rule and its application to composite functions. We identified the outer and inner functions, differentiated them separately, and then combined the results using the chain rule formula. The final result is:

dydx=3(3x−3+2x)2(2−9x−4)\frac{dy}{dx} = 3(3x^{-3} + 2x)^2 (2 - 9x^{-4})

This problem demonstrates the power and importance of the chain rule in calculus. By understanding and applying the chain rule correctly, we can differentiate a wide range of composite functions. This skill is fundamental in various applications of calculus, including optimization problems, related rates problems, and the analysis of complex mathematical models. The chain rule is not just a formula to memorize; it's a concept that allows us to break down complex differentiation problems into manageable steps. Mastering the chain rule opens the door to solving more challenging calculus problems and applying calculus to real-world scenarios.