Calculate The Derivative Of The Function.$\[ F(x) = \left(x^2 - 3x\right)^{-2} \left(6 - X^2\right)^{0.5} \\]$\[ F^{\prime}(x) = \\]

Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to its input. Calculating the derivative of a complex function can be challenging, but it is a crucial concept in mathematics and has numerous applications in various fields. In this article, we will discuss how to calculate the derivative of a complex function using the chain rule and other differentiation techniques.

The Chain Rule

The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. A composite function is a function that is defined in terms of another function. The chain rule states that if we have a composite function of the form:

${ f(x) = g(h(x)) }$

where $g$ and $h$ are functions, then the derivative of $f$ with respect to $x$ is given by:

${ f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x) }$

Applying the Chain Rule to the Given Function

The given function is:

${ f(x) = \left(x^2 - 3x\right)^{-2} \left(6 - x^2\right)^{0.5} }$

To calculate the derivative of this function, we can use the chain rule. We can rewrite the function as:

${ f(x) = g(h(x)) \cdot k(x) }$

where:

${ g(h(x)) = \left(x^2 - 3x\right)^{-2} }$

${ k(x) = \left(6 - x^2\right)^{0.5} }$

Now, we can apply the chain rule to calculate the derivative of $f$ with respect to $x$:

${ f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x) \cdot k(x) + g(h(x)) \cdot k^{\prime}(x) }$

Calculating the Derivatives of the Inner Functions

To calculate the derivative of $f$, we need to calculate the derivatives of the inner functions $g$ and $k$. We can do this using the power rule and the chain rule.

For the function $g$, we have:

${ g(h(x)) = \left(x^2 - 3x\right)^{-2} }$

Using the power rule, we can calculate the derivative of $g$ with respect to $h(x)$:

${ g^{\prime}(h(x)) = -2 \left(x^2 - 3x\right)^{-3} }$

Now, we need to calculate the derivative of $h(x)$ with respect to $x$. We can do this using the power rule:

${ h^{\prime}(x) = 2x - 3 }$

For the function $k$, we have:

${ k(x) = \left(6 - x^2\right)^{0.5} }$

Using the power rule, we can calculate the derivative of $k$ with respect to $x$:

${ k^{\prime}(x) = -x \left(6 - x^2\right)^{-0.5} }$

Calculating the Derivative of the Function

Now that we have calculated the derivatives of the inner functions, we can calculate the derivative of the function $f$ with respect to $x$:

${ f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x) \cdot k(x) + g(h(x)) \cdot k^{\prime}(x) }$

Substituting the expressions for $g^{\prime}(h(x))$, $h^{\prime}(x)$, $k(x)$, and $k^{\prime}(x)$, we get:

${ f^{\prime}(x) = -2 \left(x^2 - 3x\right)^{-3} \cdot (2x - 3) \cdot \left(6 - x^2\right)^{0.5} + \left(x^2 - 3x\right)^{-2} \cdot \left(-x \left(6 - x^2\right)^{-0.5}\right) }$

Simplifying the expression, we get:

${ f^{\prime}(x) = \frac{-4x + 6}{\left(x^2 - 3x\right)^3} \cdot \left(6 - x^2\right)^{0.5} - \frac{x}{\left(x^2 - 3x\right)^2} \cdot \left(6 - x^2\right)^{-0.5} }$

Conclusion

Calculating the derivative of a complex function can be challenging, but it is a crucial concept in mathematics and has numerous applications in various fields. In this article, we discussed how to calculate the derivative of a complex function using the chain rule and other differentiation techniques. We applied the chain rule to the given function and calculated the derivatives of the inner functions using the power rule and the chain rule. Finally, we calculated the derivative of the function with respect to $x$ and simplified the expression.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Calculus, 1st edition, Michael Spivak

Glossary

• Chain rule: A fundamental concept in calculus that allows us to differentiate composite functions.
• Composite function: A function that is defined in terms of another function.
• Power rule: A differentiation technique that states that if $f(x) = x^n$, then $f^{\prime}(x) = n x^{n-1}$.
• Derivative: A measure of the rate of change of a function with respect to its input.
  Calculating the Derivative of a Complex Function: Q&A =====================================================

Introduction

In our previous article, we discussed how to calculate the derivative of a complex function using the chain rule and other differentiation techniques. In this article, we will answer some common questions related to calculating the derivative of a complex function.

Q: What is the chain rule?

A: The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. A composite function is a function that is defined in terms of another function. The chain rule states that if we have a composite function of the form:

${ f(x) = g(h(x)) }$

where $g$ and $h$ are functions, then the derivative of $f$ with respect to $x$ is given by:

${ f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x) }$

Q: How do I apply the chain rule to a complex function?

A: To apply the chain rule to a complex function, you need to identify the inner functions and their derivatives. Then, you can use the chain rule to calculate the derivative of the outer function with respect to the inner function. Finally, you can multiply the derivative of the outer function by the derivative of the inner function to get the final derivative.

Q: What is the power rule?

A: The power rule is a differentiation technique that states that if $f(x) = x^n$, then $f^{\prime}(x) = n x^{n-1}$. This rule is used to calculate the derivative of a function that is raised to a power.

Q: How do I calculate the derivative of a function that is raised to a power?

A: To calculate the derivative of a function that is raised to a power, you can use the power rule. For example, if we have a function of the form:

${ f(x) = x^n }$

then the derivative of $f$ with respect to $x$ is given by:

${ f^{\prime}(x) = n x^{n-1} }$

Q: What is the product rule?

A: The product rule is a differentiation technique that states that if we have a function of the form:

${ f(x) = u(x) \cdot v(x) }$

then the derivative of $f$ with respect to $x$ is given by:

${ f^{\prime}(x) = u^{\prime}(x) \cdot v(x) + u(x) \cdot v^{\prime}(x) }$

Q: How do I calculate the derivative of a function that is a product of two functions?

A: To calculate the derivative of a function that is a product of two functions, you can use the product rule. For example, if we have a function of the form:

${ f(x) = u(x) \cdot v(x) }$

then the derivative of $f$ with respect to $x$ is given by:

${ f^{\prime}(x) = u^{\prime}(x) \cdot v(x) + u(x) \cdot v^{\prime}(x) }$

Q: What is the quotient rule?

A: The quotient rule is a differentiation technique that states that if we have a function of the form:

${ f(x) = \frac{u(x)}{v(x)} }$

then the derivative of $f$ with respect to $x$ is given by:

${ f^{\prime}(x) = \frac{u^{\prime}(x) \cdot v(x) - u(x) \cdot v^{\prime}(x)}{v(x)^2} }$

Q: How do I calculate the derivative of a function that is a quotient of two functions?

A: To calculate the derivative of a function that is a quotient of two functions, you can use the quotient rule. For example, if we have a function of the form:

${ f(x) = \frac{u(x)}{v(x)} }$

then the derivative of $f$ with respect to $x$ is given by:

${ f^{\prime}(x) = \frac{u^{\prime}(x) \cdot v(x) - u(x) \cdot v^{\prime}(x)}{v(x)^2} }$

Conclusion

Calculating the derivative of a complex function can be challenging, but it is a crucial concept in mathematics and has numerous applications in various fields. In this article, we answered some common questions related to calculating the derivative of a complex function. We hope that this article has been helpful in clarifying some of the concepts related to calculating the derivative of a complex function.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Calculus, 1st edition, Michael Spivak

Glossary

• Chain rule: A fundamental concept in calculus that allows us to differentiate composite functions.
• Composite function: A function that is defined in terms of another function.
• Power rule: A differentiation technique that states that if $f(x) = x^n$, then $f^{\prime}(x) = n x^{n-1}$.
• Product rule: A differentiation technique that states that if we have a function of the form $f(x) = u(x) \cdot v(x)$, then the derivative of $f$ with respect to $x$ is given by $f^{\prime}(x) = u^{\prime}(x) \cdot v(x) + u(x) \cdot v^{\prime}(x)$.
• Quotient rule: A differentiation technique that states that if we have a function of the form $f(x) = \frac{u(x)}{v(x)}$, then the derivative of $f$ with respect to $x$ is given by $f^{\prime}(x) = \frac{u^{\prime}(x) \cdot v(x) - u(x) \cdot v^{\prime}(x)}{v(x)^2}$.