Proving the General Power Rule of Differentiation: A Comprehensive Guide
Proving the General Power Rule of Differentiation: A Comprehensive Guide
The general power rule of differentiation is a fundamental concept in calculus. It states that if f(x) x^n, where n is any real number, then the derivative f'(x) is given by:
f'(x) n x^{n-1}
This article will explore how to prove this rule using several methods, including the definition of the derivative, the binomial theorem, and implicit differentiation.
Proof Using the Definition of the Derivative
Step 1: Start with the Definition of the Derivative
The definition of the derivative is:
f'(x) lim_{h to 0} (f(x h) - f(x)) / h
Step 2: Apply to f(x) x^n
For f(x) x^n, we have:
f(x h) (x h)^n
Step 3: Expand Using the Binomial Theorem
Using the binomial theorem, we expand (x h)^n as follows:
(x h)^n Σ_{k0}^{n} (binom{n}{k} x^{n-k} h^k)
Plugging this into the derivative formula:
f'(x) lim_{h to 0} ((x h)^n - x^n) / h
Expanding the numerator:
((x h)^n - x^n) Σ_{k0}^{n} (binom{n}{k} x^{n-k} h^k) - x^n
Substitute this back into the derivative formula:
f'(x) lim_{h to 0} (Σ_{k0}^{n} (binom{n}{k} x^{n-k} h^k) - x^n) / h
Factor out h from the numerator:
f'(x) lim_{h to 0} (Σ_{k1}^{n} (binom{n}{k} x^{n-k} h^{k-1}) h^n / h)
As h to 0, all terms containing h vanish, leaving:
f'(x) n x^{n-1}
Alternative Proofs
Proof Using Implicit Differentiation
Consider the equation:
y x^n
Taking the natural logarithm of both sides:
ln(y) n ln(x)
Differentiating both sides with respect to x:
(y'/y) n/x
Solving for y':
y' (ny/x) (nx^n/x) nx^{n-1}
Proof Using the Binomial Theorem for Real Numbers
The binomial theorem is generally used for integer exponents, but we can extend it to real numbers for a more general proof. For x y^p, where p is a real number, the binomial theorem is:
x y^p Σ_{k0}^{∞} (binom{p}{k} x^{p-k} y^k)
Applying this to (x h)^n:
(x h)^n Σ_{k0}^{n} (binom{n}{k} x^{n-k} h^k)
Making the substitution into the derivative formula:
lim_{h to 0} ((x h)^n - x^n) / h
Expanding the numerator:
(x h)^n - x^n Σ_{k0}^{n} (binom{n}{k} x^{n-k} h^k) - x^n
Factoring out h from the numerator:
lim_{h to 0} (Σ_{k1}^{n} (binom{n}{k} x^{n-k} h^{k-1}) h^n / h)
As h to 0, all terms containing h vanish, leaving:
f'(x) n x^{n-1}
Conclusion
Thus, we have shown that the derivative of x^n is n x^{n-1}. This proof can be applied to any real number n, thereby extending the power rule for differentiation.
Additional Insights
The binomial theorem is not just a standalone mathematical concept. It is also the foundation for the Maclaurin series of 1 / x^n. This series can be used to derive the power rule for differentiation. This approach can be seen as an extension of the concept to non-integer exponents, making it even more versatile.