Find The Derivative Of: − 9 X 4 + 3 X 8 -9x^4 + 3x^8 − 9 X 4 + 3 X 8

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 focus on finding the derivative of a given polynomial function, specifically $-9x^4 + 3x^8$.

What is a Derivative?

A derivative is a measure of how a function changes as its input changes. It is denoted by the symbol $\frac{d}{dx}$ and is calculated by taking the limit of the difference quotient. The derivative of a function $f(x)$ is denoted by $f'(x)$ and represents the rate of change of the function at a given point.

Rules of Differentiation

There are several rules of differentiation that can be used to find the derivative of a function. These rules include:

• Power Rule: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$
• Product Rule: If $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$
• Quotient Rule: If $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}$

Finding the Derivative of $-9x^4 + 3x^8$

To find the derivative of $-9x^4 + 3x^8$, we will use the power rule of differentiation. The power rule states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. We can apply this rule to each term in the given function.

Step 1: Differentiate the first term

The first term in the given function is $-9x^4$. Using the power rule, we can find its derivative as follows:

$$\frac{d}{dx}(-9x^4) = -9 \cdot 4x^{4-1} = -36x^3$$

Step 2: Differentiate the second term

The second term in the given function is $3x^8$. Using the power rule, we can find its derivative as follows:

$$\frac{d}{dx}(3x^8) = 3 \cdot 8x^{8-1} = 24x^7$$

Step 3: Combine the derivatives

Now that we have found the derivatives of each term, we can combine them to find the derivative of the given function:

$$\frac{d}{dx}(-9x^4 + 3x^8) = -36x^3 + 24x^7$$

Conclusion

In this article, we have found the derivative of the given polynomial function $-9x^4 + 3x^8$ using the power rule of differentiation. We have broken down the process into three steps and have shown how to apply the power rule to each term in the function. The derivative of the given function is $-36x^3 + 24x^7$.

Applications of Derivatives

Derivatives have numerous applications in various fields, including physics, engineering, and economics. Some of the key applications of derivatives include:

• Optimization: Derivatives can be used to find the maximum or minimum value of a function.
• Physics: Derivatives can be used to describe the motion of objects and to calculate forces and energies.
• Engineering: Derivatives can be used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Derivatives can be used to model economic systems and to make predictions about future economic trends.

Final Thoughts

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Introduction

In our previous article, we discussed the concept of derivatives and how to find the derivative of a given polynomial function. In this article, we will answer some frequently asked questions about derivatives and provide additional examples to help solidify your understanding of the concept.

Q: What is the derivative of a constant function?

A: The derivative of a constant function is 0. This is because the derivative represents the rate of change of a function, and a constant function does not change.

Q: What is the derivative of a linear function?

A: The derivative of a linear function is the slope of the line. For example, if we have a linear function $f(x) = 2x + 3$, then the derivative is $f'(x) = 2$.

Q: What is the derivative of a quadratic function?

A: The derivative of a quadratic function is a linear function. For example, if we have a quadratic function $f(x) = x^2 + 2x + 1$, then the derivative is $f'(x) = 2x + 2$.

Q: What is the derivative of a polynomial function?

A: The derivative of a polynomial function is found by differentiating each term in the polynomial. For example, if we have a polynomial function $f(x) = 3x^2 + 2x - 1$, then the derivative is $f'(x) = 6x + 2$.

Q: What is the derivative of a trigonometric function?

A: The derivative of a trigonometric function is found using the following rules:

• $\frac{d}{dx}(\sin x) = \cos x$
• $\frac{d}{dx}(\cos x) = -\sin x$
• $\frac{d}{dx}(\tan x) = \sec^2 x$

Q: What is the derivative of an exponential function?

A: The derivative of an exponential function is found using the following rule:

• $\frac{d}{dx}(e^x) = e^x$

Q: What is the derivative of a logarithmic function?

A: The derivative of a logarithmic function is found using the following rule:

• $\frac{d}{dx}(\ln x) = \frac{1}{x}$

Q: How do I find the derivative of a function with multiple variables?

A: To find the derivative of a function with multiple variables, you need to use partial derivatives. A partial derivative is a derivative of a function with respect to one of its variables, while keeping the other variables constant.

Q: What is the chain rule?

A: The chain rule is a rule for finding the derivative of a composite function. A composite function is a function that is composed of two or more functions. The chain rule states that if we have a composite function $f(g(x))$, then the derivative is $f'(g(x)) \cdot g'(x)$.

Q: What is the product rule?

A: The product rule is a rule for finding the derivative of a product of two functions. The product rule states that if we have a product of two functions $f(x) \cdot g(x)$, then the derivative is $f'(x) \cdot g(x) + f(x) \cdot g'(x)$.

Q: What is the quotient rule?

A: The quotient rule is a rule for finding the derivative of a quotient of two functions. The quotient rule states that if we have a quotient of two functions $\frac{f(x)}{g(x)}$, then the derivative is $\frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{g(x)^2}$.

Conclusion

In this article, we have answered some frequently asked questions about derivatives and provided additional examples to help solidify your understanding of the concept. Derivatives are a fundamental concept in mathematics and have numerous applications in various fields. We hope that this article has been helpful in your understanding of derivatives.