Find the derivative of the function

Oct 13, 2025 by windailys.com 125 views

Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to its input. Evaluating derivatives at specific values of x is a crucial concept in understanding the behavior of functions. In this article, we will explore how to evaluate the derivative of a given function at a specific value of x.

What is a Derivative?

A derivative is a measure of how a function changes as its input changes. It is calculated by taking the limit of the difference quotient as the change in the input approaches zero. Mathematically, the derivative of a function f(x) is denoted as f'(x) and is calculated as:

f'(x) = lim(h → 0) [f(x + h) - f(x)]/h

Evaluating Derivatives: A Step-by-Step Guide

To evaluate the derivative of a function at a specific value of x, we can follow these steps:

Find the derivative of the function: Use the power rule, product rule, or quotient rule to find the derivative of the function.
Substitute the value of x: Substitute the given value of x into the derivative.
Simplify the expression: Simplify the resulting expression to find the value of the derivative at the given value of x.

Example: Evaluating the Derivative of a Quadratic Function

Let's consider the function f(x) = -4x^2 + 7x - 5. We want to find the derivative of this function at x = 5.

Step 1: Find the Derivative of the Function

To find the derivative of the function, we can use the power rule, which states that if f(x) = x^n, then f'(x) = nx^(n-1).

Using the power rule, we can find the derivative of the function as follows:

f'(x) = d(-4x^2 + 7x - 5)/dx = -8x + 7

Step 2: Substitute the Value of x

Now that we have found the derivative of the function, we can substitute the value of x = 5 into the derivative.

f'(5) = -8(5) + 7 = -40 + 7 = -33

Conclusion

In this article, we have explored how to evaluate the derivative of a function at a specific value of x. We have used the power rule to find the derivative of a quadratic function and then substituted the value of x into the derivative to find the value of the derivative at the given value of x.

Common Mistakes to Avoid

When evaluating derivatives, there are several common mistakes to avoid:

Not using the correct rule: Make sure to use the correct rule for finding the derivative, such as the power rule, product rule, or quotient rule.
Not simplifying the expression: Make sure to simplify the resulting expression to find the value of the derivative at the given value of x.
Not checking the units: Make sure to check the units of the derivative to ensure that they are correct.

Real-World Applications

Evaluating derivatives has numerous real-world applications, including:

Physics: Derivatives are used to describe the motion of objects, including velocity and acceleration.
Engineering: Derivatives are used to optimize systems, including maximizing profit and minimizing cost.
Economics: Derivatives are used to model economic systems, including supply and demand.

Conclusion

In conclusion, evaluating derivatives is a crucial concept in calculus that has numerous real-world applications. By following the steps outlined in this article, you can evaluate the derivative of a function at a specific value of x and apply this knowledge to real-world problems.

Final Answer

The final answer is: -33

Introduction

In our previous article, we explored how to evaluate the derivative of a function at a specific value of x. In this article, we will answer some common questions related to evaluating derivatives.

Q: What is the derivative of a function?

A: The derivative of a function is a measure of how the function changes as its input changes. It is calculated by taking the limit of the difference quotient as the change in the input approaches zero.

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

A: To find the derivative of a function, you can use the power rule, product rule, or quotient rule. The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1). The product rule states that if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x). The quotient rule states that if f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2.

Q: What is the difference between the derivative and the slope of a tangent line?

A: The derivative of a function at a point represents the slope of the tangent line to the function at that point. In other words, the derivative gives the rate of change of the function at a specific point.

Q: How do I evaluate the derivative of a function at a specific value of x?

A: To evaluate the derivative of a function at a specific value of x, you can follow these steps:

Find the derivative of the function using the power rule, product rule, or quotient rule.
Substitute the value of x into the derivative.
Simplify the resulting expression to find the value of the derivative at the given value of x.

Q: What are some common mistakes to avoid when evaluating derivatives?

A: Some common mistakes to avoid when evaluating derivatives include:

Not using the correct rule for finding the derivative
Not simplifying the resulting expression
Not checking the units of the derivative

Q: What are some real-world applications of evaluating derivatives?

A: Evaluating derivatives has numerous real-world applications, including:

Physics: Derivatives are used to describe the motion of objects, including velocity and acceleration.
Engineering: Derivatives are used to optimize systems, including maximizing profit and minimizing cost.
Economics: Derivatives are used to model economic systems, including supply and demand.

Q: How do I use the chain rule to evaluate the derivative of a composite function?

A: The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x). To use the chain rule, you can follow these steps:

Identify the outer function g(x) and the inner function h(x).
Find the derivative of the outer function g(x) with respect to x.
Find the derivative of the inner function h(x) with respect to x.
Multiply the derivatives of the outer and inner functions to find the derivative of the composite function.

Q: What is the relationship between the derivative and the integral of a function?

A: The derivative and the integral of a function are inverse processes. The derivative of a function gives the rate of change of the function, while the integral of a function gives the accumulation of the function.

Conclusion

Final Answer

The final answer is: -33