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Finding the Derivative of x^4: How-To & Steps

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson we show the short way to find the derivative of x^4 and then verify this result using the limits definition of the derivative. As an application, we look at the formula describing the power radiated by the earth.

Derivative of x4

Finding the derivative using the power rule means for xn, the derivative is nxn-1. In words: n is moved in front of x and the exponent is reduced by 1 to become n - 1.

Let's find the derivative of x raised to the fourth power:


f(x)=x^4


Step 1: Focus on the exponent.

The exponent is 4.

Step 2: Make a copy of the exponent and place it in front.

If there is already a coefficient in front of x the exponent would multiply it to become the new coefficient. For f(x) = x4, the coefficient in front of x4, is 1. Multiplying 4 times 1 gives 4.

Step 3: Subtract one from the exponent.


4x^(4-1)


Step 4: Clean up the expression.


4x^3


And that's all there is to it!

Using an Alternate Way to Get the Solution

Just for fun, we verify this result using the limit definition of the derivative:


limit_definition_of_derivative


We already have f(x) so it's easy to obtain f(x + h). Wherever there is an x in f(x), replace the x with x + h. Thus, f(x) = x4 becomes f(x + h) = (x + h)4:


f(x+h)=(x+h)^4


From the limit definition, let's build and then simplify f(x + h) - f(x) divided by h:


(f(x+h)-f(x))/h=((x+h)^4-x^4)/h


The numerator has (x + h)4 which can be expanded. One way to expand this is to multiply (x + h)2 by (x + h)2 where (x + h)2 is x2 + 2xh + h2. Keeping track of common terms,

(x + h)4 = x4 + 4x3h + 6x2h2 + 4xh3 + h4.

Substituting:


f_prime(x)


The x4 cancels with the -x4, leaving:


f_prime(x)_expanded


We can divide each term in the numerator by the h in the denominator. Some of the h terms cancel:


f_prime(x)_simplified


Simplifying:


f_prime(x)_simplified_further


Take the limit as h goes to zero by substituting 0 for h on the right-hand side and then simplifying:


f_prime(x)_simplified_even_further


4x3 agrees with our earlier result.

An Application: How Much Power Does the Earth Radiate?

Φ is the amount of power radiated per unit area (watts per meter2) by an ideal radiating object. Φ depends on the temperature of the object. A very good estimate for Φ is given by the Stefan-Boltzmann equation:


Stefan-Boltzmann


σ is the Stefan-Boltzmann constant equal to 5.67x10-8 watts/(meter2 K). The temperature, T, is measured in Kelvin. This equation says if we know the temperature of an object, we can calculate how much power is radiated per square meter.

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