Online Derivative With Steps, Derivative Calculator: Wolfram

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

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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:

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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”s 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 1 from the exponent.

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As you can see in our case here, we”re subtracting 1 from our exponent of 4 to give us our new expression.

Step 4: Clean up the expression.

So there you have it! Our expression is now:

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And that”s all there is to it!

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0:04 Derivative of x^4 1:04 Using an Alternate Method 2:36 Application & Example 4:39 Lesson Summary
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Using an Alternate Method

Just for fun, let”s verify this result using the limit definition of the derivative, which you can see appearing here on your screen:

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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. Therefore, f(x) = x4 becomes f(x + h) = (x + h)4, as you can see:

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From the limit definition, let”s build and then simplify f(x + h) – f(x) divided by h, like this:

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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.

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Substituting, we get the following that you can see appearing here:

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Then the x4 cancels with the -x4, leaving the following simplified expression:

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We can divide each term in the numerator by the h in the denominator. Some of the h terms cancel as well, like you can see here:

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And now, simplifying again, we get this:

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

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So, as we can see, 4×3 agrees with our earlier result.

Application & Example

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, which states that:

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A little confusing sounding, right? Let”s break this down. σ is the Stefan-Boltzmann constant equal to 5.67 * 10-8 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.

The physics might seem complicated, but the equation is as simple as f(x) = x4. Do you see how the Φ has the role of f(x)? And T is our independent variable?

Differentiating this function uses the same steps as differentiating f(x) = x4. By the way, we might want to differentiate this science equation if we wanted to know if a change in temperature, when things are already hot, has a greater impact than when things are cold. In fact they do! But all we”ll do here is practice finding the derivative.

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Step 1: Focus on the exponent.

The exponent is 4.

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

The coefficient, σ, becomes 4σ.

Step 3: Subtract 1 from the exponent.

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4 – 1 = 3.

Step 4: Clean up the expression.

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The mean temperature of the Earth is 288 K. Thus, Φ = 5.67 * 10-8 * 2884 ≅ 390 watts/meter2. If we could harvest this energy per unit time and convert it all to electricity, then a square, one meter by one meter, would be able to power nearly four 100-watt light bulbs. This gives us an idea of how much power the hot Earth is radiating back into the atmosphere.

Lesson Summary

All right, let”s take a moment to review what we”ve learned. In this lesson, we learned that finding the derivative using the power rule means for xn, the derivative is nxn-1, and that, in words, this means n is moved in front of x and the exponent is reduced by 1 to become n – 1. And, more importantly, we learned that in order to solve this problem of finding the derivative of x4, we only need to follow four steps:

Step 1: Focus on the exponent. Step 2: Make a copy of the exponent and place it in front. Step 3: Subtract 1 from the exponent. Step 4: Clean up the expression.

It really is as simple as that!

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