Power Rule for Derivatives¶
In calculus, the power rule is a fundamental concept used to find the derivative of functions with powers (exponents). It states that if f(x) = x^n where n is a positive integer or fractional exponent, then f'(x) = n * x^(n-1). In this article, we will introduce the power rule for derivatives and explain how to apply it.
Limit Definition of Derivative¶
Before diving into the power rule, let's first review the limit definition of a derivative. Given a function f(x), its derivative, denoted as f'(x), is defined as:
f'(x) = lim (h->0) [f(x+h) - f(x)]/h
Intuitively, the derivative represents the instantaneous rate of change of the function at a given point. In other words, it tells us how much the value of the function changes as we move infinitesimally close to the point.
Power Rule for Positive Integer Powers¶
Now let's consider a function f(x) = x^n, where n is a positive integer. To find its derivative, we will apply the limit definition of a derivative:
f'(x) = lim (h->0) [(x+h)^n - x^n]/h
We can rewrite this expression as:
f'(x) = lim (h->0) [x^n * ((1 + h/x)^n - 1)]/h
Next, we apply the binomial theorem to expand (1 + h/x)^n, which gives us:
(1 + h/x)^n = 1 + n*(h/x) + ...
where the ellipsis represents higher-order terms in h. Substituting this into our expression for f'(x), we get:
f'(x) = lim (h->0) [x^n * (n*(h/x) + ...)]/h
As h approaches 0, the higher-order terms in h become negligible. Therefore, we can simplify our expression to:
f'(x) = n * x^(n-1)
So, for a function f(x) = x^n, where n is a positive integer, its derivative is given by the power rule:
f'(x) = n * x^(n-1)
Power Rule for Rational Exponents¶
Now let's extend our discussion to rational exponents. Suppose we have a function f(x) = x^m/n, where m and n are positive integers, and n is not equal to 0. To find its derivative, we will apply the quotient rule, which states that:
d/dx (u/v) = (v*du/dx - u*dv/dx)/v^2
where u(x) = x^m and v(x) = x^n. Applying this rule, we get:
f'(x) = (nx^(n-1)*dm - mx^(m-1)*dn)/x^(n+m)
Simplifying this expression gives us the power rule for rational exponents:
f'(x) = m * x^(m-1) / n - n * x^(n-1) / m
This formula can be used to find the derivative of any function with a rational exponent, as long as both m and n are positive integers and n is not equal to 0.
Examples¶
Let's now apply the power rule to several examples:
- Find the derivative of
f(x) = x^3 + 2x^2 - 3x + 7.
Applying the power rule for each term, we get:
f'(x) = d/dx (x^3) + d/dx (2x^2) - d/dx (-3x) + d/dx (7)
The derivatives of these terms are 3 * x^2, 4 * x, -3, and 0, respectively. Therefore, the derivative of f(x) = x^3 + 2x^2 - 3x + 7 is:
f'(x) = 3 * x^2 + 4 * x - 3
- Find the derivative of
g(t) = t^4/5 - 2t^3.
Applying the power rule for rational exponents, we get:
g'(t) = (4 * t^3/5 - 6 * t^2) / 1
Simplifying this expression gives us the derivative of g(t) = t^4/5 - 2t^3:
g'(t) = (4/5)*t^2 - 6*t
When the Power Rule Applies and Does Not Apply¶
The power rule applies whenever a function has an exponent that is either a positive integer or a fraction. However, it does not apply to functions with exponents that are negative integers or zero. For example, f(x) = x^-2 and g(x) = x^0 do not have derivatives according to the power rule. Instead, we use different rules for handling these cases:
- For a function with an exponent of -1, such as
h(x) = 1/x, we apply the chain rule:
h'(x) = d/dx (1/x) = -1 / x^2
- For a function with an exponent of 0, such as
k(x) = 7, its derivative is always 0:
k'(x) = d/dx (7) = 0
Conclusion¶
In this article, we introduced the power rule for derivatives and explained how to apply it to functions with positive integer powers. We then extended our discussion to include rational exponents and provided several worked examples. Finally, we highlighted when the power rule applies and does not apply. Understanding the power rule is crucial for solving problems in calculus and related fields, such as physics and engineering.