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Difficulty: MediumDifferentiation from First Principles

When differentiating the quadratic function f(x)=5x2−2xf(x) = 5x^2 - 2x from first principles, what is the fully simplified form of the difference quotient f(x+h)−f(x)h\frac{f(x+h) - f(x)}{h} before taking the limit as h→0h \to 0?

Answer: 10x + 5h - 2 / 10x - 2 + 5h / 5h + 10x - 2 / 10x+5h-2

Answer

10x+5h−210x + 5h - 2
Expanding f(x+h)=5(x+h)2−2(x+h)f(x+h) = 5(x+h)^2 - 2(x+h) yields 5x2+10xh+5h2−2x−2h5x^2 + 10xh + 5h^2 - 2x - 2h. Subtracting f(x)=5x2−2xf(x) = 5x^2 - 2x leaves 10xh+5h2−2h10xh + 5h^2 - 2h. Factoring out hh and dividing by hh gives the simplified difference quotient 10x+5h−210x + 5h - 2.

Step-by-Step Solution

1
Evaluate f(x+h)f(x+h) by expanding 5(x+h)2−2(x+h)5(x+h)^2 - 2(x+h)
f(x+h)=5(x2+2xh+h2)−2x−2h=5x2+10xh+5h2−2x−2hf(x+h) = 5(x^2 + 2xh + h^2) - 2x - 2h = 5x^2 + 10xh + 5h^2 - 2x - 2h
Substitute (x+h)(x+h) into the function definition.
2
Subtract f(x)f(x) from f(x+h)f(x+h)
f(x+h)−f(x)=(5x2+10xh+5h2−2x−2h)−(5x2−2x)=10xh+5h2−2hf(x+h) - f(x) = (5x^2 + 10xh + 5h^2 - 2x - 2h) - (5x^2 - 2x) = 10xh + 5h^2 - 2h
Determine the numerator of the difference quotient by cancelling common terms.
3
Divide the numerator by hh
\frac{f(x+h) - f(x)}{h} = \frac{10xh + 5h^2 - 2h}{h} = 10x + 5h - 2
Simplify the fraction by dividing each term by hh.

Key Concept

Difference quotient in differentiation from first principles
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Topics

Differentiation from First Principles

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