Find The Derivative Of The Function $y = 10 - 7x^2$.A) $10 - 14x$ B) $-14x$ C) $10 - 7x$ D) $-14$

Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to one of its variables. In this article, we will focus on finding the derivative of a quadratic function, specifically the function $y = 10 - 7x^2$. The derivative of this function will be used to determine the rate of change of the function at any given point.

What is a Quadratic Function?

A quadratic function is a polynomial function of degree two, which means that the highest power of the variable (in this case, $x$) is two. The general form of a quadratic function is $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In the function $y = 10 - 7x^2$, the coefficient of $x^2$ is $-7$, the coefficient of $x$ is $0$, and the constant term is $10$.

The Power Rule of Differentiation

To find the derivative of a quadratic function, we will use the power rule of differentiation. The power rule states that if $y = x^n$, then $y' = nx^{n-1}$. This rule can be applied to any term in a polynomial function, as long as the exponent is a positive integer.

Applying the Power Rule to the Function

To find the derivative of the function $y = 10 - 7x^2$, we will apply the power rule to the term $-7x^2$. Using the power rule, we get:

$$\frac{d}{dx}(-7x^2) = -7(2)x^{2-1} = -14x$$

Finding the Derivative of the Constant Term

The derivative of a constant term is always zero. In this case, the constant term is $10$, so the derivative of $10$ is $0$.

Combining the Derivatives of the Terms

Now that we have found the derivatives of the individual terms, we can combine them to find the derivative of the entire function. The derivative of the function $y = 10 - 7x^2$ is:

$$y' = -14x + 0 = -14x$$

Conclusion

In this article, we found the derivative of the quadratic function $y = 10 - 7x^2$ using the power rule of differentiation. The derivative of this function is $y' = -14x$. This result can be used to determine the rate of change of the function at any given point.

Answer

The correct answer is B) $-14x$.

Step-by-Step Solution

1. Identify the function: $y = 10 - 7x^2$
2. Apply the power rule to the term $-7x^2$: $\frac{d}{dx}(-7x^2) = -7(2)x^{2-1} = -14x$
3. Find the derivative of the constant term: $\frac{d}{dx}(10) = 0$
4. Combine the derivatives of the terms: $y' = -14x + 0 = -14x$

Tips and Tricks

• When applying the power rule, make sure to multiply the coefficient by the exponent.
• The derivative of a constant term is always zero.
• When combining the derivatives of the terms, make sure to add or subtract the derivatives correctly.

Common Mistakes

• Forgetting to multiply the coefficient by the exponent when applying the power rule.
• Not recognizing that the derivative of a constant term is always zero.
• Not combining the derivatives of the terms correctly.

Real-World Applications

• The derivative of a function can be used to determine the rate of change of the function at any given point.
• The derivative of a function can be used to find the maximum or minimum value of the function.
• The derivative of a function can be used to solve optimization problems.

Practice Problems

• Find the derivative of the function $y = 3x^2 - 2x + 1$.
• Find the derivative of the function $y = 2x^3 + 5x^2 - 3x + 1$.
• Find the derivative of the function $y = x^4 - 2x^3 + 3x^2 - x + 1$.

Conclusion

In this article, we found the derivative of the quadratic function $y = 10 - 7x^2$ using the power rule of differentiation. The derivative of this function is $y' = -14x$. This result can be used to determine the rate of change of the function at any given point. We also discussed the power rule of differentiation, the derivative of a constant term, and how to combine the derivatives of the terms. Finally, we provided some practice problems for the reader to try.

Introduction

In our previous article, we discussed how to find the derivative of a quadratic function using the power rule of differentiation. In this article, we will answer some common questions related to finding the derivative of a quadratic function.

Q: What is the derivative of the function $y = x^2 + 3x - 2$?

A: To find the derivative of the function $y = x^2 + 3x - 2$, we will apply the power rule to each term. The derivative of $x^2$ is $2x$, the derivative of $3x$ is $3$, and the derivative of $-2$ is $0$. Therefore, the derivative of the function is $y' = 2x + 3$.

Q: How do I find the derivative of a quadratic function with a negative coefficient?

A: To find the derivative of a quadratic function with a negative coefficient, you can apply the power rule as usual. For example, if you have the function $y = -x^2 + 3x - 2$, you can apply the power rule to each term to get $y' = -2x + 3$.

Q: Can I use the power rule to find the derivative of a function with a fractional exponent?

A: No, the power rule can only be used to find the derivative of a function with a positive integer exponent. If you have a function with a fractional exponent, you will need to use a different method to find the derivative.

Q: How do I find the derivative of a function with a constant term?

A: The derivative of a constant term is always zero. For example, if you have the function $y = 3x^2 + 5x - 2$, the derivative of the constant term $-2$ is $0$.

Q: Can I use the power rule to find the derivative of a function with a variable in the exponent?

A: No, the power rule can only be used to find the derivative of a function with a constant exponent. If you have a function with a variable in the exponent, you will need to use a different method to find the derivative.

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

A: To find the derivative of a function with multiple terms, you can apply the power rule to each term separately and then combine the results. For example, if you have the function $y = x^2 + 3x - 2$, you can apply the power rule to each term to get $y' = 2x + 3$.

Q: Can I use the power rule to find the derivative of a function with a trigonometric function?

A: No, the power rule can only be used to find the derivative of a function with a polynomial function. If you have a function with a trigonometric function, you will need to use a different method to find the derivative.

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

A: To find the derivative of a function with a logarithmic function, you can use the chain rule and the power rule. For example, if you have the function $y = \ln(x^2 + 3x - 2)$, you can use the chain rule and the power rule to get $y' = \frac{2x + 3}{x^2 + 3x - 2}$.

Q: Can I use the power rule to find the derivative of a function with a rational function?

A: No, the power rule can only be used to find the derivative of a function with a polynomial function. If you have a function with a rational function, you will need to use a different method to find the derivative.

Conclusion

In this article, we answered some common questions related to finding the derivative of a quadratic function. We discussed how to find the derivative of a function with a negative coefficient, a fractional exponent, a constant term, and multiple terms. We also discussed how to find the derivative of a function with a trigonometric function, a logarithmic function, and a rational function.