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ChatGPT
May 19
Differentiate means to recognize or point out a difference between two or more things. It can also refer to the process of finding the derivative of a function in calculus.
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Google Gemini AI
May 19
Differentiation is a mathematical operation that finds the rate of change of a function with respect to one or more of its variables. It is used to calculate the slope of a curve, the velocity of an object, and the acceleration due to gravity.
To differentiate a function, you use the following rules:
- Power rule: If f(x) = x^n, then f'(x) = nx^(n-1)
- Product rule: If f(x) = g(x)h(x), then f'(x) = g'(x)h(x) + g(x)h'(x)
- Quotient rule: If f(x) = g(x)/h(x), then f'(x) = (h(x)g'(x) - g(x)h'(x))/(h(x)^2)
- Chain rule: If f(x) = g(h(x)), then f'(x) = g'(h(x))h'(x)
Example:
Find the derivative of f(x) = x^3 + 2x^2 - 5x + 1.
Solution:
Using the power rule, we have:
f'(x) = 3x^2 + 4x - 5
To differentiate a function, you use the following rules:
- Power rule: If f(x) = x^n, then f'(x) = nx^(n-1)
- Product rule: If f(x) = g(x)h(x), then f'(x) = g'(x)h(x) + g(x)h'(x)
- Quotient rule: If f(x) = g(x)/h(x), then f'(x) = (h(x)g'(x) - g(x)h'(x))/(h(x)^2)
- Chain rule: If f(x) = g(h(x)), then f'(x) = g'(h(x))h'(x)
Example:
Find the derivative of f(x) = x^3 + 2x^2 - 5x + 1.
Solution:
Using the power rule, we have:
f'(x) = 3x^2 + 4x - 5
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