Find The Derivative Of The Function $f(x$\] Below. It May Be To Your Advantage To Simplify First.$f(x) = \left(x^6 - \sqrt[5]{x}\right) 7^x$f^{\prime}(x) =$

Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to its input. It is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and economics. In this article, we will explore the process of finding the derivative of a complex function, specifically the function $f(x) = \left(x^6 - \sqrt[5]{x}\right) 7^x$. We will break down the function into simpler components, apply the rules of differentiation, and finally, obtain the derivative of the function.

Understanding the Function

The given function is $f(x) = \left(x^6 - \sqrt[5]{x}\right) 7^x$. This function consists of two main components: a polynomial expression $x^6 - \sqrt[5]{x}$ and an exponential expression $7^x$. To find the derivative of the function, we need to apply the product rule of differentiation, which states that if we have a function of the form $f(x) = u(x)v(x)$, then the derivative of the function is given by $f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$.

Simplifying the Function

Before we apply the product rule, let's simplify the function by rewriting the radical expression $\sqrt[5]{x}$ as $x^{1/5}$. This gives us $f(x) = \left(x^6 - x^{1/5}\right) 7^x$. Now, we can see that the function consists of two terms: $x^6$ and $-x^{1/5}$, both multiplied by $7^x$.

Applying the Product Rule

To find the derivative of the function, we will apply the product rule. Let $u(x) = x^6 - x^{1/5}$ and $v(x) = 7^x$. Then, we have:

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

Finding the Derivative of u(x)

To find the derivative of $u(x) = x^6 - x^{1/5}$, we will apply the power rule of differentiation, which states that if we have a function of the form $f(x) = x^n$, then the derivative of the function is given by $f^{\prime}(x) = nx^{n-1}$. Applying this rule to the two terms in $u(x)$, we get:

$$u^{\prime}(x) = 6x^5 - \frac{1}{5}x^{-4/5}$$

Finding the Derivative of v(x)

To find the derivative of $v(x) = 7^x$, we will apply the chain rule of differentiation, which states that if we have a function of the form $f(x) = g(h(x))$, then the derivative of the function is given by $f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x)$. In this case, we have $g(x) = 7^x$ and $h(x) = x$. Then, we have:

$$v^{\prime}(x) = 7^x \ln(7)$$

Substituting the Derivatives

Now that we have found the derivatives of $u(x)$ and $v(x)$, we can substitute them into the product rule formula:

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

$$f^{\prime}(x) = \left(6x^5 - \frac{1}{5}x^{-4/5}\right) 7^x + \left(x^6 - x^{1/5}\right) 7^x \ln(7)$$

Simplifying the Derivative

To simplify the derivative, we can combine like terms:

$$f^{\prime}(x) = 6x^5 7^x - \frac{1}{5}x^{-4/5} 7^x + x^6 7^x \ln(7) - x^{1/5} 7^x \ln(7)$$

Final Answer

The derivative of the function $f(x) = \left(x^6 - \sqrt[5]{x}\right) 7^x$ is given by:

$$f^{\prime}(x) = 6x^5 7^x - \frac{1}{5}x^{-4/5} 7^x + x^6 7^x \ln(7) - x^{1/5} 7^x \ln(7)$$

Introduction

In our previous article, we explored the process of finding the derivative of a complex function, specifically the function $f(x) = \left(x^6 - \sqrt[5]{x}\right) 7^x$. We broke down the function into simpler components, applied the rules of differentiation, and finally, obtained the derivative of the function. In this article, we will address some common questions and concerns related to the derivative of a complex function.

Q: What is the derivative of a complex function?

A: The derivative of a complex function is a measure of how the function changes as its input changes. It is a fundamental concept in calculus and has numerous applications in various fields, including physics, engineering, and economics.

Q: How do I find the derivative of a complex function?

A: To find the derivative of a complex function, you need to break down the function into simpler components, apply the rules of differentiation, and finally, obtain the derivative of the function. This may involve using the product rule, chain rule, and power rule of differentiation.

Q: What is the product rule of differentiation?

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

Q: What is the chain rule of differentiation?

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

Q: How do I apply the product rule and chain rule of differentiation?

A: To apply the product rule and chain rule of differentiation, you need to identify the components of the function and apply the corresponding rules of differentiation. For example, if you have a function of the form $f(x) = u(x)v(x)$, you can apply the product rule by finding the derivatives of $u(x)$ and $v(x)$ and substituting them into the product rule formula.

Q: What is the power rule of differentiation?

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

Q: How do I apply the power rule of differentiation?

A: To apply the power rule of differentiation, you need to identify the exponent of the function and multiply it by the coefficient of the function. For example, if you have a function of the form $f(x) = x^3$, you can apply the power rule by multiplying the exponent by the coefficient, resulting in $f^{\prime}(x) = 3x^2$.

Q: What are some common mistakes to avoid when finding the derivative of a complex function?

A: Some common mistakes to avoid when finding the derivative of a complex function include:

• Failing to break down the function into simpler components
• Applying the wrong rule of differentiation
• Failing to substitute the derivatives of the components into the product rule or chain rule formula
• Failing to simplify the derivative

Conclusion

In conclusion, finding the derivative of a complex function requires a thorough understanding of the rules of differentiation, including the product rule, chain rule, and power rule. By breaking down the function into simpler components, applying the corresponding rules of differentiation, and finally, obtaining the derivative of the function, you can successfully find the derivative of a complex function.