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Calculate derivatives of polynomial functions with step-by-step solutions. Apply the power rule, constant rule, and sum rule with detailed working shown.
A derivative is a fundamental concept in calculus that measures the instantaneous rate of change of a function, representing the slope of the tangent line at any given point.
Use x as the variable, ^ for exponents (e.g., x^3 + 2x - 1)
| f(x) | f'(x) | Rule |
|---|---|---|
| x^n | nx^(n-1) | Power Rule |
| sin(x) | cos(x) | Trig |
| cos(x) | -sin(x) | Trig |
| tan(x) | sec²(x) | Trig |
| e^x | e^x | Exponential |
| ln(x) | 1/x | Logarithmic |
| a^x | a^x · ln(a) | Exponential |
Formula
d/dx[x^n] = n·x^(n-1) (Power Rule)x = The independent variable
n = The exponent (power) of x
d/dx = The differentiation operator with respect to x
Worked Example
Differentiate f(x) = 4x³ + 2x² - 5x + 3
Did you know? Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus in the late 17th century, leading to one of the most famous priority disputes in the history of mathematics (source: Stanford Encyclopedia of Philosophy).
Sources
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