Master the core principles of differentiation, including rules and visual animations to explain rates of change.
Start CalculusDerivatives measure how a function changes as its input changes. They are fundamental in physics (velocity, acceleration), economics (marginal revenue), and optimization problems.
General Derivative Formula:
f'(x) = limₕ→0 [f(x+h) - f(x)] / h
Slope of tangent line to f(x) = x² at x=2
d/dx [xⁿ] = n·x^(n-1)
Example: d/dx [x³] = 3x²
d/dx [uv] = u'v + uv'
Example: d/dx [x·sinx] = sinx + x·cosx
d/dx [f(g(x))] = f'(g(x))·g'(x)
Example: d/dx [sin(2x)] = 2·cos(2x)
What is the derivative of 3x⁴ - 5x + 7?
Steps: Derivative of 3x⁴ = 12x³ | Derivative of -5x = -5 | Derivative of constant = 0
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