Introduction and what you need to know
In this article I will be showing you how to prove a famous rule in calculus known as the product rule. It can be proven in several ways. The one I am using is the most practical. To understand this article you need to have a basic knowledge of implicit differentiation.
The proof
The proof is actually much easier than you may think. Suppose you have an exponent x^n.
y = x^n
Take the natural logarithm of both sides.
log y = log x^n
According to the laws of logarithms that is equivalent to log y = n log x
Now take the derivative of both sides using implicit differentiation. Treat n as a constant.
(dy/dx) * 1/y = n/x
Multiply both sides by y.
(dy/dx) = ny/x
We know that y = x^n, so plug that in for y.
nx^n/x
Dividing nx^n by x yields nx^(n - 1) by exponent laws. That is what the power rule states. f'(x^n) = nx^(n - 1). You now know how to prove one of the most useful and easy to apply rules in calculus. Congratulations!
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