Calculating Derivatives Of Composite Functions F'(a) And G'(a)

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Introduction

In this article, we delve into the fascinating world of calculus, specifically focusing on the computation of derivatives for composite functions. We are given two functions, $F(x)$ and $G(x)$, defined in terms of another function $f(x)$. Our mission is to determine the derivatives of these composite functions, $F'(x)$ and $G'(x)$, at a specific point. This exploration will require a solid understanding of the chain rule, a fundamental concept in differential calculus. We will also leverage given values of the function $f(x)$, its derivative $f'(x)$, and a specific constant $a$ to arrive at our solutions. The problem not only tests our ability to apply the chain rule but also our algebraic manipulation skills, making it a comprehensive exercise in calculus.

Defining the Functions and Given Values

Let's first formally define the functions we'll be working with. We have $F(x) = f(x^7)$ and $G(x) = (f(x))^7$. These are composite functions, where $F(x)$ is the function $f$ applied to $x^7$, and $G(x)$ is the function $f(x)$ raised to the power of 7. Additionally, we are provided with the following crucial information:

  • a6=15a^6 = 15

  • f(a)=3f(a) = 3

  • f′(a)=4f'(a) = 4

  • f′(a7)=14f'(a^7) = 14

These values will serve as the foundation for our calculations. We will use them in conjunction with the chain rule to find $F'(a)$ and $G'(a)$.

Calculating $F'(a)$

To find $F'(a)$, we first need to determine the derivative of $F(x)$ with respect to $x$, which is $F'(x)$. Since $F(x) = f(x^7)$, we will employ the chain rule. The chain rule states that if we have a composite function $h(x) = f(g(x))$, then its derivative is given by $h'(x) = f'(g(x)) \cdot g'(x)$.

Applying the Chain Rule

In our case, $g(x) = x^7$, so we need to find the derivative of $g(x)$, which is $g'(x) = 7x^6$. Now, applying the chain rule to $F(x)$, we get:

F′(x)=f′(x7)imes(7x6)F'(x) = f'(x^7) imes (7x^6)

This expression gives us the derivative of $F(x)$ at any point $x$. To find $F'(a)$, we substitute $x = a$ into the equation:

F′(a)=f′(a7)imes(7a6)F'(a) = f'(a^7) imes (7a^6)

Substituting Known Values

We are given that $f'(a^7) = 14$ and $a^6 = 15$. Substituting these values into the expression for $F'(a)$, we have:

F′(a)=14imes(7imes15)F'(a) = 14 imes (7 imes 15)

Final Calculation for $F'(a)$

Now, we simply perform the arithmetic:

F′(a)=14imes105=1470F'(a) = 14 imes 105 = 1470

Therefore, the value of $F'(a)$ is 1470. This result is a direct application of the chain rule combined with the given values, showcasing the power of calculus in solving intricate problems involving composite functions.

Calculating $G'(a)$

Now, let's shift our focus to finding $G'(a)$. Recall that $G(x) = (f(x))^7$. To determine $G'(a)$, we first need to find the derivative of $G(x)$ with respect to $x$, denoted as $G'(x)$. Similar to the previous calculation, we will utilize the chain rule, but this time, the composition is slightly different.

Applying the Chain Rule to $G(x)$

Here, we can consider the outer function as $h(u) = u^7$, where $u = f(x)$ is the inner function. The derivative of $h(u)$ with respect to $u$ is $h'(u) = 7u^6$, and the derivative of $f(x)$ with respect to $x$ is $f'(x)$. Applying the chain rule, we get:

G′(x)=7(f(x))6imesf′(x)G'(x) = 7(f(x))^6 imes f'(x)

This equation represents the derivative of $G(x)$ at any point $x$. To find $G'(a)$, we substitute $x = a$ into the equation:

G′(a)=7(f(a))6imesf′(a)G'(a) = 7(f(a))^6 imes f'(a)

Substituting Known Values

We are given that $f(a) = 3$ and $f'(a) = 4$. Substituting these values into the expression for $G'(a)$, we have:

G′(a)=7(3)6imes4G'(a) = 7(3)^6 imes 4

Final Calculation for $G'(a)$

Now, let's perform the arithmetic:

G′(a)=7imes729imes4G'(a) = 7 imes 729 imes 4

G′(a)=7imes2916G'(a) = 7 imes 2916

G′(a)=20412G'(a) = 20412

Thus, the value of $G'(a)$ is 20412. This calculation further demonstrates the application of the chain rule in conjunction with given values to find the derivative of a composite function.

Conclusion

In this article, we successfully calculated the derivatives of two composite functions, $F(x) = f(x^7)$ and $G(x) = (f(x))^7$, at a specific point $a$. We found that $F'(a) = 1470$ and $G'(a) = 20412$. These calculations hinged on a solid understanding and application of the chain rule, a cornerstone of differential calculus. Additionally, we effectively utilized given values of the function $f(x)$, its derivative $f'(x)$, and the constant $a$ to arrive at our solutions. This exercise underscores the importance of mastering fundamental calculus concepts and their application in solving complex problems.

Key Takeaways

  • The chain rule is essential for finding the derivatives of composite functions.
  • Careful substitution of known values is crucial for accurate calculations.
  • Understanding the structure of composite functions is vital for applying the chain rule correctly.
  • Calculus provides powerful tools for analyzing and solving problems involving rates of change.

By working through this problem, we have reinforced our understanding of these key principles and enhanced our problem-solving skills in calculus. This knowledge will undoubtedly be valuable in tackling future challenges in mathematics and related fields.