Roger Saanye
Feb 6
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Feb 6
Differential calculus is a branch of calculus that deals with the study of rates of change of functions. It focuses on finding the derivative of a function, which represents the rate at which the function is changing at any given point.
Here are a few examples to illustrate the concept of differential calculus:
1. Example: Find the derivative of the function f(x) = 3x^2 + 2x - 1.
Solution: To find the derivative, we differentiate each term of the function separately. The derivative of 3x^2 is 6x (using the power rule), the derivative of 2x is 2 (using the constant rule), and the derivative of -1 is 0 (as the derivative of a constant is always 0). Therefore, the derivative of f(x) is f'(x) = 6x + 2.
2. Example: Find the derivative of the function g(x) = sin(x) + cos(x).
Solution: To find the derivative, we differentiate each term of the function separately. The derivative of sin(x) is cos(x) (using the derivative of sine rule), and the derivative of cos(x) is -sin(x) (using the derivative of cosine rule). Therefore, the derivative of g(x) is g'(x) = cos(x) - sin(x).
3. Example: Find the derivative of the function h(x) = e^x.
Solution: The derivative of e^x is simply e^x (using the derivative of exponential rule). Therefore, the derivative of h(x) is h'(x) = e^x.
These examples demonstrate how differential calculus allows us to find the rate of change of various functions, which is useful in many fields such as physics, economics, and engineering.
Here are a few examples to illustrate the concept of differential calculus:
1. Example: Find the derivative of the function f(x) = 3x^2 + 2x - 1.
Solution: To find the derivative, we differentiate each term of the function separately. The derivative of 3x^2 is 6x (using the power rule), the derivative of 2x is 2 (using the constant rule), and the derivative of -1 is 0 (as the derivative of a constant is always 0). Therefore, the derivative of f(x) is f'(x) = 6x + 2.
2. Example: Find the derivative of the function g(x) = sin(x) + cos(x).
Solution: To find the derivative, we differentiate each term of the function separately. The derivative of sin(x) is cos(x) (using the derivative of sine rule), and the derivative of cos(x) is -sin(x) (using the derivative of cosine rule). Therefore, the derivative of g(x) is g'(x) = cos(x) - sin(x).
3. Example: Find the derivative of the function h(x) = e^x.
Solution: The derivative of e^x is simply e^x (using the derivative of exponential rule). Therefore, the derivative of h(x) is h'(x) = e^x.
These examples demonstrate how differential calculus allows us to find the rate of change of various functions, which is useful in many fields such as physics, economics, and engineering.
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