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Difficulty: MediumDifferentiation from First Principles

Given the function f(x)=5x2−4x+3f(x) = 5x^2 - 4x + 3, what is the numerical value of its derivative at x=2x = 2 when evaluated using the first principles limit definition lim⁡h→0f(x+h)−f(x)h\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}?

Answer: 16

Answer

The numerical value of the derivative of f(x)=5x2−4x+3f(x) = 5x^2 - 4x + 3 at x=2x = 2 is 16.
Evaluating the first principles definition lim⁡h→0f(x+h)−f(x)h\lim_{h \to 0} \frac{f(x+h) - f(x)}{h} for f(x)=5x2−4x+3f(x) = 5x^2 - 4x + 3 yields the derivative f′(x)=10x−4f'(x) = 10x - 4. Substituting x=2x = 2 yields 10(2)−4=1610(2) - 4 = 16.

Step-by-Step Solution

1
Determine the expanded form of f(x+h)f(x+h)
f(x+h)=5(x+h)2−4(x+h)+3=5x2+10xh+5h2−4x−4h+3f(x+h) = 5(x+h)^2 - 4(x+h) + 3 = 5x^2 + 10xh + 5h^2 - 4x - 4h + 3
Evaluating the function at x+hx+h requires expanding the square and distributing the constant factors.
2
Compute the difference f(x+h)−f(x)f(x+h) - f(x)
f(x+h)−f(x)=(5x2+10xh+5h2−4x−4h+3)−(5x2−4x+3)=10xh+5h2−4hf(x+h) - f(x) = (5x^2 + 10xh + 5h^2 - 4x - 4h + 3) - (5x^2 - 4x + 3) = 10xh + 5h^2 - 4h
Subtracting f(x)f(x) cancels terms independent of hh.
3
Form and simplify the difference quotient f(x+h)−f(x)h\frac{f(x+h) - f(x)}{h}
10xh+5h2−4hh=10x+5h−4\frac{10xh + 5h^2 - 4h}{h} = 10x + 5h - 4
Factoring hh out of the numerator allows division by hh for non-zero hh.
4
Evaluate the limit as h→0h \to 0
f′(x)=lim⁡h→0(10x+5h−4)=10x−4f'(x) = \lim_{h \to 0} (10x + 5h - 4) = 10x - 4
Taking the limit produces the general derivative function dydx\frac{\mathrm{d}y}{\mathrm{d}x}.
5
Substitute x=2x = 2 into the derivative
f′(2)=10(2)−4=16f'(2) = 10(2) - 4 = 16
Evaluating at x=2x = 2 gives the instantaneous rate of change at that point.

Key Concept

Differentiation from First Principles
Estimated Time:1m 30s
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Differentiation from First Principles

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