The Quotient Rule of Differentiation

Oct 13, 2025 by miodeya.com 75 views

Introduction

In this article, we will be evaluating the derivative of a given function $f(x)$ at a specific point $x=-\pi$ . The function $f(x)$ is defined as $f(x)=\frac{-11 \cdot x}{\sin (x)+\cos (x)}$ . To evaluate the derivative of this function, we will use the quotient rule of differentiation, which states that if we have a function of the form $f(x)=\frac{g(x)}{h(x)}$ , then the derivative of this function is given by $f^{\prime}(x)=\frac{h(x)g^{\prime}(x)-g(x)h^{\prime}(x)}{(h(x))^2}$ .

The Quotient Rule of Differentiation

The quotient rule of differentiation is a powerful tool that allows us to find the derivative of a function that is defined as the ratio of two other functions. This rule is essential in calculus and is used to find the derivative of many functions that are defined as ratios.

To apply the quotient rule, we need to find the derivatives of the numerator and denominator of the function. In this case, the numerator is $-11x$ and the denominator is $\sin(x)+\cos(x)$ . We will use the power rule of differentiation to find the derivative of the numerator and the sum rule of differentiation to find the derivative of the denominator.

Finding the Derivative of the Numerator

The derivative of the numerator $-11x$ is found using the power rule of differentiation, which states that if we have a function of the form $f(x)=x^n$ , then the derivative of this function is given by $f^{\prime}(x)=nx^{n-1}$ . In this case, the derivative of $-11x$ is $-11$ .

Finding the Derivative of the Denominator

The derivative of the denominator $\sin(x)+\cos(x)$ is found using the sum rule of differentiation, which states that if we have a function of the form $f(x)=g(x)+h(x)$ , then the derivative of this function is given by $f^{\prime}(x)=g^{\prime}(x)+h^{\prime}(x)$ . In this case, the derivative of $\sin(x)+\cos(x)$ is $\cos(x)-\sin(x)$ .

Applying the Quotient Rule

Now that we have found the derivatives of the numerator and denominator, we can apply the quotient rule to find the derivative of the function $f(x)$ . The quotient rule states that if we have a function of the form $f(x)=\frac{g(x)}{h(x)}$ , then the derivative of this function is given by $f^{\prime}(x)=\frac{h(x)g^{\prime}(x)-g(x)h^{\prime}(x)}{(h(x))^2}$ .

Substituting the derivatives of the numerator and denominator into this formula, we get:

$f^{\prime}(x)=\frac{(\sin(x)+\cos(x))(-11)-(-11x)(\cos(x)-\sin(x))}{(\sin(x)+\cos(x))^2}$

Simplifying the Derivative

To simplify the derivative, we can expand the numerator and denominator and then combine like terms.

Expanding the numerator, we get:

$-11\sin(x)-11\cos(x)+11x\cos(x)-11x\sin(x)$

Expanding the denominator, we get:

$(\sin(x)+\cos(x))^2=\sin^2(x)+2\sin(x)\cos(x)+\cos^2(x)$

Combining like terms in the numerator, we get:

$-11\sin(x)-11\cos(x)+11x\cos(x)-11x\sin(x)=-11(\sin(x)+\cos(x))+11x(\cos(x)-\sin(x))$

Evaluating the Derivative at $x=-\pi$

Now that we have simplified the derivative, we can evaluate it at $x=-\pi$ . Substituting $x=-\pi$ into the derivative, we get:

$f^{\prime}(-\pi)=\frac{(-\pi)(\cos(-\pi)+\sin(-\pi))-11(\cos(-\pi)+\sin(-\pi))}{(\cos(-\pi)+\sin(-\pi))^2}$

Using Trigonometric Identities

To evaluate the derivative at $x=-\pi$ , we can use trigonometric identities to simplify the expression.

Recall that $\cos(-\pi)=-1$ and $\sin(-\pi)=0$ . Substituting these values into the derivative, we get:

$f^{\prime}(-\pi)=\frac{(-\pi)(-1+0)-11(-1+0)}{(-1+0)^2}$

Simplifying the Expression

To simplify the expression, we can evaluate the numerator and denominator separately.

Evaluating the numerator, we get:

$-\pi+11$

Evaluating the denominator, we get:

$1$

Final Answer

Therefore, the final answer is:

$f^{\prime}(-\pi)=\frac{-\pi+11}{1}=-\pi+11$

Conclusion

In this article, we evaluated the derivative of a given function $f(x)$ at a specific point $x=-\pi$ . We used the quotient rule of differentiation to find the derivative of the function and then simplified the expression using trigonometric identities. The final answer is $-\pi+11$ .

Introduction

In our previous article, we evaluated the derivative of the function $f(x)=\frac{-11 \cdot x}{\sin (x)+\cos (x)}$ at $x=-\pi$ . We used the quotient rule of differentiation to find the derivative of the function and then simplified the expression using trigonometric identities. In this article, we will answer some common questions related to this topic.

Q: What is the quotient rule of differentiation?

A: The quotient rule of differentiation is a rule that allows us to find the derivative of a function that is defined as the ratio of two other functions. It states that if we have a function of the form $f(x)=\frac{g(x)}{h(x)}$ , then the derivative of this function is given by $f^{\prime}(x)=\frac{h(x)g^{\prime}(x)-g(x)h^{\prime}(x)}{(h(x))^2}$ .

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

A: To apply the quotient rule, you need to find the derivatives of the numerator and denominator of the function. Then, you can substitute these derivatives into the quotient rule formula and simplify the expression.

Q: What is the power rule of differentiation?

A: The power rule of differentiation is a rule that allows us to find the derivative of a function that is defined as a power of x. It states that if we have a function of the form $f(x)=x^n$ , then the derivative of this function is given by $f^{\prime}(x)=nx^{n-1}$ .

Q: What is the sum rule of differentiation?

A: The sum rule of differentiation is a rule that allows us to find the derivative of a function that is defined as the sum of two other functions. It states that if we have a function of the form $f(x)=g(x)+h(x)$ , then the derivative of this function is given by $f^{\prime}(x)=g^{\prime}(x)+h^{\prime}(x)$ .

Q: How do I simplify the derivative of a function?

A: To simplify the derivative of a function, you can use various techniques such as combining like terms, factoring, and using trigonometric identities.

Q: What is the final answer to the problem of evaluating the derivative of $f(x)=\frac{-11 \cdot x}{\sin (x)+\cos (x)}$ at $x=-\pi$ ?

A: The final answer is $-\pi+11$ .

Q: Can I use the quotient rule to find the derivative of any function?

A: No, the quotient rule can only be used to find the derivative of a function that is defined as the ratio of two other functions.

Q: What are some common mistakes to avoid when applying the quotient rule?

A: Some common mistakes to avoid when applying the quotient rule include:

Forgetting to find the derivatives of the numerator and denominator
Not substituting the derivatives into the quotient rule formula
Not simplifying the expression after applying the quotient rule

Conclusion

In this article, we answered some common questions related to evaluating the derivative of a function using the quotient rule. We also provided some tips and tricks for simplifying the derivative of a function. We hope that this article has been helpful in clarifying any confusion you may have had about this topic.