The Derivative as a Limit (Difference Quotient) (Calculus I)
Free printable university Calculus I worksheet: computing derivatives from the formal definition f'(x) = lim [f(x+h) - f(x)]/h for quadratics, reciprocals and square roots. Every answer cross-checked numerically.
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Calculus I · Math worksheet
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The Derivative as a Limit (Difference Quotient)
Differentiate from first principles: form the difference quotient [f(x + h) - f(x)]/h, simplify it algebraically until h no longer appears in a denominator, then let h approach 0.
- 1.Use the definition f'(x) = lim (h -> 0) of [f(x + h) - f(x)] / h to differentiate f(x) = -3x^2.
- 2.Use the definition f'(x) = lim (h -> 0) of [f(x + h) - f(x)] / h to differentiate f(x) = -2x^2 - 2x + 6.
- 3.Use the definition f'(x) = lim (h -> 0) of [f(x + h) - f(x)] / h to differentiate f(x) = + 6x + 4.
- 4.Use the definition f'(x) = lim (h -> 0) of [f(x + h) - f(x)] / h to differentiate f(x) = - + 3x - 6.
- 5.Use the definition of the derivative to differentiate f(x) = 1/(x - 2).
- 6.Use the definition of the derivative to differentiate f(x) = 1/(x + 3).
- 7.For f(x) = √x, use the definition of the derivative to find f'(36).
- 8.For f(x) = √x, use the definition of the derivative to find f'(49).
- 9.For f(x) = -2x^2 - x + 4, use the definition f'(-2) = lim (h -> 0) of [f(-2 + h) - f(-2)] / h to find f'(-2).
- 10.For f(x) = -2x^2 + 4x + 2, use the definition f'(-1) = lim (h -> 0) of [f(-1 + h) - f(-1)] / h to find f'(-1).
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