Understanding the Power Rule: A Fundamental Tool for Calculus Differentiation

Power Rule

The power rule is a fundamental rule in calculus that allows us to differentiate functions that involve powers of a variable

The power rule is a fundamental rule in calculus that allows us to differentiate functions that involve powers of a variable.

The power rule states that if we have a function of the form f(x) = x^n, where n is a constant exponent, the derivative of this function is given by:

f'(x) = nx^(n-1)

In other words, to differentiate a function with a power exponent, we bring down the exponent as a coefficient and decrease the exponent by 1.

Let’s look at some examples to understand how to apply the power rule:

Example 1:
Find the derivative of f(x) = x^3.

We can apply the power rule here by bringing down the exponent as a coefficient and decreasing the exponent by 1:

f'(x) = 3x^(3-1)
= 3x^2

So, the derivative of f(x) = x^3 is f'(x) = 3x^2.

Example 2:
Find the derivative of g(x) = 5x^4.

Again, we can apply the power rule where the constant 5 is multiplied by the derivative of x^4:

g'(x) = 5 * 4x^(4-1)
= 20x^3

So, the derivative of g(x) = 5x^4 is g'(x) = 20x^3.

Example 3:
Find the derivative of h(x) = √x (square root of x).

In this case, we can rewrite the square root as a fractional exponent:

h(x) = x^(1/2)

Now, applying the power rule, we have:

h'(x) = (1/2)x^((1/2)-1)
= (1/2)x^(-1/2)
= 1/(2√x)

So, the derivative of h(x) = √x is h'(x) = 1/(2√x).

The power rule is a very useful tool in calculus as it allows us to easily find derivatives of functions involving powers. However, it is important to note that the power rule only applies when the exponent is a constant. If the exponent contains a variable, we need to use more advanced rules, such as the product rule or chain rule, to find the derivative.

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