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Difficulty: HardDifferentiation from First Principles

Using the first principles of differentiation for the reciprocal function f(x)=4xf(x) = \frac{4}{x}, evaluate the limit of the difference quotient as h→0h \to 0: lim⁡h→0f(x+h)−f(x)h\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}. What is the simplified expression for the derivative dydx\frac{dy}{dx}?

Answer: -\frac{4}{x^2} / -4/x^2 / -4 / x^2 / -\frac{4}{x^{2}}

Answer

The derivative dydx\frac{dy}{dx} is −4x2-\frac{4}{x^2}.
Substituting f(x)=4xf(x) = \frac{4}{x} into the first principles limit formula yields lim⁡h→04x−4(x+h)h⋅x(x+h)=lim⁡h→0−4hh⋅x(x+h)\lim_{h \to 0} \frac{4x - 4(x+h)}{h \cdot x(x+h)} = \lim_{h \to 0} \frac{-4h}{h \cdot x(x+h)}. Canceling hh gives lim⁡h→0−4x(x+h)=−4x2\lim_{h \to 0} -\frac{4}{x(x+h)} = -\frac{4}{x^2}.

Step-by-Step Solution

1
Write the first principles formula and substitute f(x)=4xf(x) = \frac{4}{x}.
\frac{dy}{dx} = \lim_{h \to 0} \frac{\frac{4}{x+h} - \frac{4}{x}}{h}
Apply the definition of differentiation from first principles.
2
Combine the fractions in the numerator using a common denominator x(x+h)x(x+h).
\frac{4}{x+h} - \frac{4}{x} = \frac{4x - 4(x+h)}{x(x+h)} = \frac{4x - 4x - 4h}{x(x+h)} = \frac{-4h}{x(x+h)}
Simplify the numerator into a single fractional expression.
3
Divide the simplified numerator by hh and cancel the common factor of hh.
\frac{\frac{-4h}{x(x+h)}}{h} = \frac{-4h}{h \cdot x(x+h)} = -\frac{4}{x(x+h)}
Eliminate the indeterminate factor of hh from the denominator.
4
Evaluate the limit as h→0h \to 0.
\lim_{h \to 0} -\frac{4}{x(x+h)} = -\frac{4}{x(x+0)} = -\frac{4}{x^2}
Substitute h=0h = 0 into the simplified algebraic expression.

Key Concept

Differentiation from First Principles
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Topics

Differentiation from First Principles

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