Basic Derivative
A derivative is a fundamental concept in calculus that represents the rate of change of a function with respect to its independent variable
A derivative is a fundamental concept in calculus that represents the rate of change of a function with respect to its independent variable. It measures how a function changes as the input variable changes. The derivative of a function f(x) is denoted by f'(x), df(x)/dx, or dy/dx.
The basic derivative, or the derivative of a basic function, refers to finding the derivative of a simple elementary function. These basic functions include:
1. Constant Function: The derivative of a constant function f(x) = c, where c is a constant, is always zero.
Example: If f(x) = 5, then f'(x) = 0.
2. Linear Function: The derivative of a linear function f(x) = mx + b, where m is the slope, and b is the y-intercept, is equal to the slope (m).
Example: If f(x) = 2x + 3, then f'(x) = 2.
3. Power Rule: The derivative of a power function f(x) = x^n, where n is a constant, is given by multiplying the exponent by x raised to the power of (n-1).
Example: If f(x) = x^3, then f'(x) = 3x^2.
4. Exponential Function: The derivative of an exponential function f(x) = a^x, where ‘a’ is a constant, is given by multiplying the function by the natural logarithm of the base (ln(a)).
Example: If f(x) = 2^x, then f'(x) = 2^x * ln(2).
5. Trigonometric Function: The derivatives of trigonometric functions involve various rules depending on the specific function.
– The derivative of sin(x) is cos(x).
– The derivative of cos(x) is -sin(x).
– The derivative of tan(x) is sec^2(x).
– The derivative of csc(x) is -csc(x) * cot(x).
– The derivative of sec(x) is sec(x) * tan(x).
– The derivative of cot(x) is -csc^2(x).
These basic derivatives serve as building blocks for more complex derivatives and are essential in various areas of calculus and higher-level mathematics. Calculating derivatives helps analyze the behavior of functions, determine slopes of tangent lines, find maximum and minimum points, and solve optimization problems, among other applications.
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