Given The Function H ( X ) = ( 6 X − 6 ) 4 Sin ⁡ 3 ( X ) H(x)=\frac{(6x-6)^4}{\sin^3(x)} H ( X ) = S I N 3 ( X ) ( 6 X − 6 ) 4 ​ , Find H ′ ( X H^{\prime}(x H ′ ( X ] In Any Form.

Introduction

In calculus, differentiation is a fundamental concept that deals with the study of rates of change and slopes of curves. Given a function, we can find its derivative, which represents the rate of change of the function with respect to its input variable. In this article, we will explore the differentiation of a complex function, specifically the function $h(x)=\frac{(6x-6)^4}{\sin^3(x)}$. Our goal is to find the derivative of this function, $h^{\prime}(x)$, in any form.

The Function

The given function is $h(x)=\frac{(6x-6)^4}{\sin^3(x)}$. This function involves a polynomial in the numerator and a trigonometric function in the denominator. To find the derivative of this function, we will apply the quotient rule and the chain rule of differentiation.

Applying the Quotient Rule

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

$$f^{\prime}(x)=\frac{h(x)g^{\prime}(x)-g(x)h^{\prime}(x)}{h(x)^2}$$

In our case, we have $g(x)=(6x-6)^4$ and $h(x)=\sin^3(x)$. We will first find the derivatives of $g(x)$ and $h(x)$ separately.

Derivative of the Numerator

To find the derivative of the numerator, we will apply the power rule of differentiation, which states that if $f(x)=x^n$, then $f^{\prime}(x)=nx^{n-1}$. In this case, we have $g(x)=(6x-6)^4$, so we can write:

$$g^{\prime}(x)=4(6x-6)^3\cdot6$$

Simplifying this expression, we get:

$$g^{\prime}(x)=24(6x-6)^3$$

Derivative of the Denominator

To find the derivative of the denominator, we will apply the chain rule of differentiation, which states that if $f(x)=g(h(x))$, then $f^{\prime}(x)=g^{\prime}(h(x))\cdot h^{\prime}(x)$. In this case, we have $h(x)=\sin^3(x)$, so we can write:

$$h^{\prime}(x)=3\sin^2(x)\cdot\cos(x)$$

Applying the Quotient Rule

Now that we have found the derivatives of the numerator and the denominator, we can apply the quotient rule to find the derivative of the function $h(x)$:

$$h^{\prime}(x)=\frac{\sin^3(x)\cdot24(6x-6)^3-24(6x-6)^3\cdot3\sin^2(x)\cos(x)}{\sin^6(x)}$$

Simplifying this expression, we get:

$$h^{\prime}(x)=\frac{24(6x-6)^3\sin^3(x)(1-3\cos^2(x))}{\sin^6(x)}$$

Simplifying the Derivative

We can simplify the derivative further by canceling out the common factors:

$$h^{\prime}(x)=\frac{24(6x-6)^3\sin(x)(1-3\cos^2(x))}{\sin^3(x)}$$

Conclusion

In this article, we have found the derivative of the complex function $h(x)=\frac{(6x-6)^4}{\sin^3(x)}$ using the quotient rule and the chain rule of differentiation. The derivative of this function is given by:

$$h^{\prime}(x)=\frac{24(6x-6)^3\sin(x)(1-3\cos^2(x))}{\sin^3(x)}$$

$$$$

Introduction

In our previous article, we explored the differentiation of a complex function, specifically the function $h(x)=\frac{(6x-6)^4}{\sin^3(x)}$. We found the derivative of this function using the quotient rule and the chain rule of differentiation. In this article, we will answer some common questions related to the differentiation of complex functions.

Q: What is the quotient rule of differentiation?

A: The quotient rule of differentiation is a formula that allows us to find the derivative of a function of the form $f(x)=\frac{g(x)}{h(x)}$. The quotient rule states that:

$$f^{\prime}(x)=\frac{h(x)g^{\prime}(x)-g(x)h^{\prime}(x)}{h(x)^2}$$

Q: How do I apply the quotient rule?

A: To apply the quotient rule, you need to follow these steps:

1. Identify the numerator and the denominator of the function.
2. Find the derivatives of the numerator and the denominator separately.
3. Plug the derivatives into the quotient rule formula.
4. Simplify the resulting expression.

Q: What is the chain rule of differentiation?

A: The chain rule of differentiation is a formula that allows us to find the derivative of a composite function. A composite function is a function of the form $f(x)=g(h(x))$. The chain rule states that:

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

Q: How do I apply the chain rule?

A: To apply the chain rule, you need to follow these steps:

1. Identify the outer function and the inner function.
2. Find the derivative of the outer function with respect to the inner function.
3. Multiply the derivative of the outer function by the derivative of the inner function.
4. Simplify the resulting expression.

Q: What is the power rule of differentiation?

A: The power rule of differentiation is a formula that allows us to find the derivative of a function of the form $f(x)=x^n$. The power rule states that:

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

Q: How do I apply the power rule?

A: To apply the power rule, you need to follow these steps:

1. Identify the exponent of the function.
2. Multiply the exponent by the coefficient of the function.
3. Subtract 1 from the exponent.
4. Write the resulting expression as the derivative of the function.

Q: What are some common mistakes to avoid when differentiating complex functions?

A: Some common mistakes to avoid when differentiating complex functions include:

• Forgetting to apply the quotient rule or the chain rule.
• Not simplifying the resulting expression.
• Making errors when finding the derivatives of the numerator and the denominator.
• Not checking the domain of the function.

Conclusion

In this article, we have answered some common questions related to the differentiation of complex functions. We have also provided some tips and tricks for applying the quotient rule, the chain rule, and the power rule of differentiation. By following these tips and tricks, you can avoid common mistakes and find the derivatives of complex functions with ease.