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Difficulty: EasyDifferentiation from First Principles

When differentiating the function f(x)=x2+4xf(x) = x^2 + 4x from first principles, what is the simplified expression for 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: 2x + h + 4 / 2x + 4 + h / h + 2x + 4

Answer

2x+h+42x + h + 4
Expanding f(x+h)=(x+h)2+4(x+h)=x2+2xh+h2+4x+4hf(x+h) = (x+h)^2 + 4(x+h) = x^2 + 2xh + h^2 + 4x + 4h. Subtracting f(x)=x2+4xf(x) = x^2 + 4x yields 2xh+h2+4h2xh + h^2 + 4h. Dividing every term by hh gives the simplified difference quotient 2x+h+42x + h + 4.

Step-by-Step Solution

1
Evaluate f(x+h)f(x+h) for the function f(x)=x2+4xf(x) = x^2 + 4x
f(x+h)=(x+h)2+4(x+h)=x2+2xh+h2+4x+4hf(x+h) = (x+h)^2 + 4(x+h) = x^2 + 2xh + h^2 + 4x + 4h
Substitute (x+h)(x+h) into every instance of xx in the function and expand the terms.
2
Subtract f(x)f(x) from f(x+h)f(x+h) to find the numerator of the difference quotient
f(x+h)−f(x)=(x2+2xh+h2+4x+4h)−(x2+4x)=2xh+h2+4hf(x+h) - f(x) = (x^2 + 2xh + h^2 + 4x + 4h) - (x^2 + 4x) = 2xh + h^2 + 4h
Subtracting original function terms cancels out x2x^2 and 4x4x.
3
Divide the result by hh
\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 + 4h}{h} = 2x + h + 4
Factor out hh from the numerator and cancel with the denominator hh.

Key Concept

Difference Quotient Simplification in First Principles Differentiation
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Topics

Differentiation from First Principles

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