Differentiate:${ Y = \ln \left(5x^2 - 7x + 6\right) }$
Differentiate: y = ln(5x^2 - 7x + 6)
In this article, we will delve into the world of calculus and explore the concept of differentiation. Differentiation is a fundamental concept in mathematics that deals with the study of rates of change and slopes of curves. It is a crucial tool in various fields such as physics, engineering, and economics. In this discussion, we will focus on differentiating the given function y = ln(5x^2 - 7x + 6).
Understanding the Function
The given function is a natural logarithmic function, which is defined as y = ln(u), where u is a function of x. In this case, the function u is given by 5x^2 - 7x + 6. To differentiate the given function, we need to apply the chain rule of differentiation, which states that if y = f(u) and u = g(x), then y' = f'(u) * g'(x).
Applying the Chain Rule
To differentiate the given function, we need to find the derivative of the natural logarithmic function and the derivative of the function u. The derivative of the natural logarithmic function is given by:
d/dx (ln(u)) = 1/u * du/dx
The derivative of the function u is given by:
du/dx = d/dx (5x^2 - 7x + 6) = 10x - 7
Now, we can substitute the value of du/dx into the equation for the derivative of the natural logarithmic function:
d/dx (ln(5x^2 - 7x + 6)) = 1/(5x^2 - 7x + 6) * (10x - 7)
Simplifying the Derivative
To simplify the derivative, we can cancel out the common factor of (10x - 7) from the numerator and denominator:
d/dx (ln(5x^2 - 7x + 6)) = (10x - 7)/(5x^2 - 7x + 6)
In this article, we have differentiated the given function y = ln(5x^2 - 7x + 6) using the chain rule of differentiation. We have found that the derivative of the function is given by (10x - 7)/(5x^2 - 7x + 6). This derivative can be used to find the rate of change of the function and the slope of the curve at any point.
To illustrate the concept of differentiation, let's consider a few example problems.
Example 1
Find the derivative of the function y = ln(x^2 + 1).
Solution
To differentiate the function, we need to apply the chain rule of differentiation. The derivative of the natural logarithmic function is given by:
d/dx (ln(u)) = 1/u * du/dx
The derivative of the function u is given by:
du/dx = d/dx (x^2 + 1) = 2x
Now, we can substitute the value of du/dx into the equation for the derivative of the natural logarithmic function:
d/dx (ln(x^2 + 1)) = 1/(x^2 + 1) * (2x) = 2x/(x^2 + 1)
Example 2
Find the derivative of the function y = ln(2x^2 - 3x + 1).
Solution
To differentiate the function, we need to apply the chain rule of differentiation. The derivative of the natural logarithmic function is given by:
d/dx (ln(u)) = 1/u * du/dx
The derivative of the function u is given by:
du/dx = d/dx (2x^2 - 3x + 1) = 4x - 3
Now, we can substitute the value of du/dx into the equation for the derivative of the natural logarithmic function:
d/dx (ln(2x^2 - 3x + 1)) = 1/(2x^2 - 3x + 1) * (4x - 3) = (4x - 3)/(2x^2 - 3x + 1)
Differentiation has numerous applications in various fields such as physics, engineering, and economics. Some of the key applications of differentiation include:
- Physics: Differentiation is used to describe the motion of objects and the forces acting on them. For example, the derivative of the position function can be used to find the velocity and acceleration of an object.
- Engineering: Differentiation is used to design and optimize systems such as electrical circuits, mechanical systems, and control systems.
- Economics: Differentiation is used to model the behavior of economic systems and to make predictions about future trends.
In this article, we have discussed the concept of differentiation and its applications in various fields. We have also differentiated the given function y = ln(5x^2 - 7x + 6) using the chain rule of differentiation. The derivative of the function is given by (10x - 7)/(5x^2 - 7x + 6). This derivative can be used to find the rate of change of the function and the slope of the curve at any point.
Differentiate: y = ln(5x^2 - 7x + 6) - Q&A
In our previous article, we discussed the concept of differentiation and its applications in various fields. We also differentiated the given function y = ln(5x^2 - 7x + 6) using the chain rule of differentiation. In this article, we will answer some frequently asked questions related to differentiation and provide additional examples to illustrate the concept.
Q: What is differentiation?
A: Differentiation is a mathematical process that deals with the study of rates of change and slopes of curves. It is a crucial tool in various fields such as physics, engineering, and economics.
Q: What is the chain rule of differentiation?
A: The chain rule of differentiation is a fundamental concept in calculus that states that if y = f(u) and u = g(x), then y' = f'(u) * g'(x). This rule is used to differentiate composite functions.
Q: How do I differentiate a natural logarithmic function?
A: To differentiate a natural logarithmic function, you need to apply the chain rule of differentiation. The derivative of the natural logarithmic function is given by:
d/dx (ln(u)) = 1/u * du/dx
Q: What is the derivative of the function y = ln(5x^2 - 7x + 6)?
A: The derivative of the function y = ln(5x^2 - 7x + 6) is given by:
d/dx (ln(5x^2 - 7x + 6)) = (10x - 7)/(5x^2 - 7x + 6)
Q: Can you provide more examples of differentiating natural logarithmic functions?
A: Yes, here are a few more examples:
Example 1
Find the derivative of the function y = ln(x^2 + 1).
Solution
To differentiate the function, we need to apply the chain rule of differentiation. The derivative of the natural logarithmic function is given by:
d/dx (ln(u)) = 1/u * du/dx
The derivative of the function u is given by:
du/dx = d/dx (x^2 + 1) = 2x
Now, we can substitute the value of du/dx into the equation for the derivative of the natural logarithmic function:
d/dx (ln(x^2 + 1)) = 1/(x^2 + 1) * (2x) = 2x/(x^2 + 1)
Example 2
Find the derivative of the function y = ln(2x^2 - 3x + 1).
Solution
To differentiate the function, we need to apply the chain rule of differentiation. The derivative of the natural logarithmic function is given by:
d/dx (ln(u)) = 1/u * du/dx
The derivative of the function u is given by:
du/dx = d/dx (2x^2 - 3x + 1) = 4x - 3
Now, we can substitute the value of du/dx into the equation for the derivative of the natural logarithmic function:
d/dx (ln(2x^2 - 3x + 1)) = 1/(2x^2 - 3x + 1) * (4x - 3) = (4x - 3)/(2x^2 - 3x + 1)
Q: What are some real-world applications of differentiation?
A: Differentiation has numerous applications in various fields such as physics, engineering, and economics. Some of the key applications of differentiation include:
- Physics: Differentiation is used to describe the motion of objects and the forces acting on them. For example, the derivative of the position function can be used to find the velocity and acceleration of an object.
- Engineering: Differentiation is used to design and optimize systems such as electrical circuits, mechanical systems, and control systems.
- Economics: Differentiation is used to model the behavior of economic systems and to make predictions about future trends.
In this article, we have answered some frequently asked questions related to differentiation and provided additional examples to illustrate the concept. We have also discussed the real-world applications of differentiation in various fields.