Use The General Power Rule To Find The Derivative Of The Function.$f(x)=\left(10 X^3-6 X\right)^{4 / 3}$f^{\prime}(x)=$

Introduction

In calculus, the general power rule is a fundamental concept used to find the derivative of a function. It states that if we have a function of the form $f(x) = (g(x))^n$, where $g(x)$ is a differentiable function and $n$ is a constant, then the derivative of $f(x)$ is given by $f^{\prime}(x) = n(g(x))^{n-1} \cdot g^{\prime}(x)$. In this article, we will use the general power rule to find the derivative of the function $f(x) = \left(10x^3 - 6x\right)^{4/3}$.

Understanding the General Power Rule

The general power rule is a powerful tool in calculus that allows us to find the derivative of a function that is raised to a power. The rule states that if we have a function of the form $f(x) = (g(x))^n$, then the derivative of $f(x)$ is given by $f^{\prime}(x) = n(g(x))^{n-1} \cdot g^{\prime}(x)$. This rule can be applied to a wide range of functions, including polynomial functions, trigonometric functions, and exponential functions.

Applying the General Power Rule to the Given Function

To find the derivative of the function $f(x) = \left(10x^3 - 6x\right)^{4/3}$, we can use the general power rule. First, we need to identify the inner function $g(x)$ and the exponent $n$. In this case, the inner function is $g(x) = 10x^3 - 6x$ and the exponent is $n = 4/3$.

Next, we need to find the derivative of the inner function $g(x)$. Using the power rule, we have $g^{\prime}(x) = 30x^2 - 6$.

Now, we can apply the general power rule to find the derivative of the function $f(x)$. We have:

$f^{\prime}(x) = \frac{4}{3} \left(10x^3 - 6x\right)^{4/3 - 1} \cdot (30x^2 - 6)$

Simplifying the expression, we get:

$f^{\prime}(x) = \frac{4}{3} \left(10x^3 - 6x\right)^{1/3} \cdot (30x^2 - 6)$

Simplifying the Derivative

To simplify the derivative, we can expand the expression using the distributive property. We have:

$f^{\prime}(x) = \frac{4}{3} \left(10x^3 - 6x\right)^{1/3} \cdot (30x^2 - 6)$

$f^{\prime}(x) = \frac{4}{3} \left(10x^3 - 6x\right)^{1/3} \cdot 30x^2 - \frac{4}{3} \left(10x^3 - 6x\right)^{1/3} \cdot 6$

Simplifying further, we get:

$f^{\prime}(x) = \frac{4}{3} \left(10x^3 - 6x\right)^{1/3} \cdot 30x^2 - \frac{8}{3} \left(10x^3 - 6x\right)^{1/3}$

Conclusion

In this article, we used the general power rule to find the derivative of the function $f(x) = \left(10x^3 - 6x\right)^{4/3}$. We identified the inner function $g(x)$ and the exponent $n$, and then applied the general power rule to find the derivative of the function. We simplified the derivative using the distributive property and obtained the final expression for the derivative.

Final Answer

The final answer is:

Introduction

In our previous article, we used the general power rule to find the derivative of the function $f(x) = \left(10x^3 - 6x\right)^{4/3}$. In this article, we will answer some common questions related to the general power rule and provide additional examples to help you understand the concept better.

Q: What is the general power rule?

A: The general power rule is a fundamental concept in calculus that allows us to find the derivative of a function that is raised to a power. The rule states that if we have a function of the form $f(x) = (g(x))^n$, then the derivative of $f(x)$ is given by $f^{\prime}(x) = n(g(x))^{n-1} \cdot g^{\prime}(x)$.

Q: How do I apply the general power rule?

A: To apply the general power rule, you need to identify the inner function $g(x)$ and the exponent $n$. Then, you need to find the derivative of the inner function $g(x)$ using the power rule. Finally, you can apply the general power rule to find the derivative of the function $f(x)$.

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

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

• Not identifying the inner function $g(x)$ and the exponent $n$
• Not finding the derivative of the inner function $g(x)$ using the power rule
• Not applying the general power rule correctly to find the derivative of the function $f(x)$

Q: Can I use the general power rule to find the derivative of a function that has a negative exponent?

A: Yes, you can use the general power rule to find the derivative of a function that has a negative exponent. However, you need to be careful when applying the rule, as the derivative of a function with a negative exponent may not be what you expect.

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

A: To simplify the derivative of a function that is raised to a power, you can use the distributive property to expand the expression. You can also use algebraic manipulations to simplify the expression further.

Q: What are some examples of functions that can be differentiated using the general power rule?

A: Some examples of functions that can be differentiated using the general power rule include:

• $f(x) = (x^2 + 1)^3$
• $f(x) = (2x - 1)^4$
• $f(x) = (x^3 + 2x^2 - 3x + 1)^2$

Q: Can I use the general power rule to find the derivative of a function that has a variable exponent?

A: Yes, you can use the general power rule to find the derivative of a function that has a variable exponent. However, you need to be careful when applying the rule, as the derivative of a function with a variable exponent may not be what you expect.

Conclusion

In this article, we answered some common questions related to the general power rule and provided additional examples to help you understand the concept better. We hope that this article has been helpful in clarifying any doubts you may have had about the general power rule.

Final Answer

The final answer is:

• The general power rule is a fundamental concept in calculus that allows us to find the derivative of a function that is raised to a power.
• To apply the general power rule, you need to identify the inner function $g(x)$ and the exponent $n$, find the derivative of the inner function $g(x)$ using the power rule, and then apply the general power rule to find the derivative of the function $f(x)$.
• Some common mistakes to avoid when applying the general power rule include not identifying the inner function $g(x)$ and the exponent $n$, not finding the derivative of the inner function $g(x)$ using the power rule, and not applying the general power rule correctly to find the derivative of the function $f(x)$.
• You can use the general power rule to find the derivative of a function that has a negative exponent, but you need to be careful when applying the rule.
• To simplify the derivative of a function that is raised to a power, you can use the distributive property to expand the expression and algebraic manipulations to simplify the expression further.
• Some examples of functions that can be differentiated using the general power rule include $f(x) = (x^2 + 1)^3$, $f(x) = (2x - 1)^4$, and $f(x) = (x^3 + 2x^2 - 3x + 1)^2$.
• You can use the general power rule to find the derivative of a function that has a variable exponent, but you need to be careful when applying the rule.