Derivatives
The power rule, the derivative at a point, tangent lines, and the chain, product and quotient rules.
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About Derivatives
The derivative is the slope of a curve at a point, and f′(x) is a formula for that slope at every x. The power rule handles any power of x: bring the exponent down in front and take 1 off it. A constant on its own has derivative 0.
The chain rule handles a function inside another, the product rule a product, and the quotient rule a fraction. The tangent line at x = a is the line through (a, f(a)) with slope f′(a).
Keep in mind
- Power rule: d/dx of xⁿ = nxⁿ⁻¹. Constants drop to 0.
- Write 1/x² as x⁻² and √x as x^(1/2) before using the power rule.
- Chain rule: derivative of the outside (inside left alone) × derivative of the inside.
- Product rule: f′g + fg′. Quotient rule: (low · d-high − high · d-low) ÷ low².
- Tangent line at x = a: y − f(a) = f′(a)(x − a).
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