Basic Derivative
The derivative is a fundamental concept in calculus that measures the rate of change of a function at a given point
The derivative is a fundamental concept in calculus that measures the rate of change of a function at a given point. It tells us how the function is changing with respect to its input variable. The derivative of a function f(x) is denoted as f'(x) or dy/dx, and it represents the slope of the tangent line to the graph of the function at a particular point.
To find the derivative of a function, there are various techniques and rules that can be used depending on the complexity of the function. However, for basic polynomials and elementary functions, there are a few general rules that simplify the process.
1. Power Rule:
The power rule is used for functions of the form f(x) = x^n, where n is a constant. The derivative of this function is given by:
f'(x) = n * x^(n-1)
For example, if f(x) = x^3, then f'(x) = 3x^(3-1) = 3x^2.
2. Constant Rule:
The derivative of a constant function is always zero. For example, if f(x) = 5, then f'(x) = 0.
3. Sum/Difference Rule:
The derivative of a sum or difference of two functions f(x) and g(x) is equal to the sum or difference of their derivatives. For example, if f(x) = x^2 + 3x and g(x) = 5x, then the derivative of f(x) – g(x) is given by:
(f – g)'(x) = f'(x) – g'(x) = (2x + 3) – 5 = 2x – 2.
4. Product Rule:
The derivative of the product of two functions f(x) and g(x) is given by:
(f * g)'(x) = f'(x) * g(x) + f(x) * g'(x)
5. Quotient Rule:
The derivative of the quotient of two functions f(x) and g(x) is given by:
(f / g)'(x) = (f'(x) * g(x) – f(x) * g'(x)) / g(x)^2
These are just a few basic rules for finding derivatives. As functions become more complicated, such as trigonometric, logarithmic, or exponential functions, additional rules and techniques are needed. It’s important to practice and understand these basic rules before moving on to more advanced topics in calculus.
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