Given The Following Table Of Values, Find $h^{\prime}(1)$ If $h(x)=\frac{5}{x^2}+\frac{f(x)}{g(x)}$. \[ \begin{tabular}{|c|c|c|c|c|} \hline X$ & F ( X ) F(x) F ( X ) & F ′ ( X ) F^{\prime}(x) F ′ ( X ) & G ( X ) G(x) G ( X ) & G ′ ( X ) G^{\prime}(x) G ′ ( X ) \ \hline 1 & 2 & -2 & -2 &

Finding the Derivative of a Complex Function

In calculus, finding the derivative of a function is a crucial concept that helps us understand the rate of change of the function with respect to its input. However, when dealing with complex functions, it can be challenging to find the derivative. In this article, we will explore how to find the derivative of a complex function using the given table of values.

The Given Function

The given function is $h(x)=\frac{5}{x^2}+\frac{f(x)}{g(x)}$. To find the derivative of this function, we need to apply the quotient rule and the chain rule of differentiation.

The Quotient Rule

The quotient rule states that if we have a function of the form $h(x)=\frac{f(x)}{g(x)}$, then the derivative of $h(x)$ is given by $h^{{\prime}(x)=\frac{f}{\prime}(x)g(x)-f(x)g^{{\prime}(x)}{(g(x))}2}$.

Applying the Quotient Rule

In our case, we have $h(x)=\frac{5}{x^2}+\frac{f(x)}{g(x)}$. We can rewrite this function as $h(x)=\frac{5g(x)}{x^2g(x)}+\frac{f(x)}{g(x)}$. Now, we can apply the quotient rule to find the derivative of $h(x)$.

Finding the Derivative of the First Term

The derivative of the first term is given by $\frac{d}{dx}\left(\frac{5g(x)}{x^{2g(x)}\right)=\frac{(x}2g(x))^{{\prime}(5g(x))-5g(x)(x}2g(x))^{\prime}}{(x2g(x))^2}$. Using the product rule, we get $\frac{d}{dx}\left(\frac{5g(x)}{x^{2g(x)}\right)=\frac{(x}2)^{{\prime}5g(x)+x}2(5g(x))^{{\prime}-5g(x)(x}2)^{{\prime}-5g(x)(g(x))}{\prime}}{(x^2g(x))2}$.

Simplifying the Derivative

Simplifying the derivative, we get $\frac{d}{dx}\left(\frac{5g(x)}{x^{2g(x)}\right)=\frac{-2x5g(x)+x}2(5g(x))^{{\prime}-5g(x)(x}2)^{{\prime}-5g(x)(g(x))}{\prime}}{(x^2g(x))2}$.

Finding the Derivative of the Second Term

The derivative of the second term is given by $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{(g(x))^{{\prime}f(x)-f(x)(g(x))}{\prime}}{(g(x))^2}$.

Combining the Derivatives

Now, we can combine the derivatives of the two terms to find the derivative of $h(x)$.

The Final Derivative

The final derivative of $h(x)$ is given by $h^{{\prime}(x)=\frac{-10g(x)}{x}3g(x)^2}+\frac{g{\prime}(x)f(x)-f(x)g^{{\prime}(x)}{g(x)}2}$.

Finding the Value of $h^{\prime}(1)$

To find the value of $h^{\prime}(1)$, we need to substitute $x=1$ into the final derivative.

Substituting $x=1$

Substituting $x=1$ into the final derivative, we get $h^{{\prime}(1)=\frac{-10g(1)}{1}3g(1)^2}+\frac{g{\prime}(1)f(1)-f(1)g^{{\prime}(1)}{g(1)}2}$.

Using the Given Table of Values

Using the given table of values, we get $h^{{\prime}(1)=\frac{-10(-2)}{1}3(-2)^{2}+\frac{(-2)(2)-2(-2)}{(-2)}2}$.

Simplifying the Expression

Simplifying the expression, we get $h^{\prime}(1)=\frac{20}{4}+\frac{-4+4}{4}$.

The Final Answer

The final answer is $h^{\prime}(1)=5+0$.

Conclusion

In this article, we explored how to find the derivative of a complex function using the given table of values. We applied the quotient rule and the chain rule of differentiation to find the derivative of the function. Finally, we used the given table of values to find the value of $h^{\prime}(1)$.
Q&A: Finding the Derivative of a Complex Function

In our previous article, we explored how to find the derivative of a complex function using the given table of values. We applied the quotient rule and the chain rule of differentiation to find the derivative of the function. In this article, we will answer some frequently asked questions related to finding the derivative of a complex function.

Q: What is the quotient rule of differentiation?

A: The quotient rule of differentiation states that if we have a function of the form $h(x)=\frac{f(x)}{g(x)}$, then the derivative of $h(x)$ is given by $h^{{\prime}(x)=\frac{f}{\prime}(x)g(x)-f(x)g^{{\prime}(x)}{(g(x))}2}$.

Q: How do I apply the quotient rule to find the derivative of a complex function?

A: To apply the quotient rule, you need to identify the numerator and denominator of the function. Then, you need to find the derivatives of the numerator and denominator separately. Finally, you can substitute the derivatives into the quotient rule formula to find the derivative of the function.

Q: What is the chain rule of differentiation?

A: The chain rule of differentiation states that if we have a function of the form $h(x)=f(g(x))$, then the derivative of $h(x)$ is given by $h^{{\prime}(x)=f}{\prime}(g(x))g^{\prime}(x)$.

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

A: To apply the chain rule, you need to identify the inner and outer functions of the function. Then, you need to find the derivatives of the inner and outer functions separately. Finally, you can substitute the derivatives into the chain rule formula to find the derivative of the function.

Q: What is the final derivative of a complex function?

A: The final derivative of a complex function is given by $h^{{\prime}(x)=\frac{-10g(x)}{x}3g(x)^2}+\frac{g{\prime}(x)f(x)-f(x)g^{{\prime}(x)}{g(x)}2}$.

Q: How do I find the value of $h^{\prime}(1)$?

A: To find the value of $h^{\prime}(1)$, you need to substitute $x=1$ into the final derivative. Then, you can use the given table of values to find the value of $h^{\prime}(1)$.

Q: What is the final answer for $h^{\prime}(1)$?

A: The final answer for $h^{\prime}(1)$ is $h^{\prime}(1)=5+0$.

Conclusion

In this article, we answered some frequently asked questions related to finding the derivative of a complex function. We provided step-by-step explanations and examples to help you understand the concepts. We hope this article has been helpful in clarifying any doubts you may have had about finding the derivative of a complex function.

Additional Resources

If you want to learn more about finding the derivative of a complex function, we recommend checking out the following resources:

• Calculus textbooks: There are many excellent calculus textbooks that cover the topic of finding the derivative of a complex function.
• Online tutorials: There are many online tutorials and videos that provide step-by-step explanations and examples of finding the derivative of a complex function.
• Practice problems: Practice problems are an excellent way to reinforce your understanding of finding the derivative of a complex function. You can find practice problems in calculus textbooks or online.

Final Thoughts

Finding the derivative of a complex function can be challenging, but with practice and patience, you can master this skill. Remember to apply the quotient rule and the chain rule of differentiation to find the derivative of a complex function. With the right tools and resources, you can become proficient in finding the derivative of a complex function.