Finding Antiderivatives
In Exercises 1–16, find an antid...

Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

2x

Accepted Answer

Welcome back everyone. Find an inside derivative for the function 4 X cubed. A 4 x 4 + C. B 16 X to the power of 4 + C, C X to the power of 4 + C, and D 16 X cubed plus C. For this problem we want to find the inside derivative F of X, such that the derivative of it is equal to 4 X cubed. So here we have to think in reverse. On the right hand side we have a polynomial function, and what we can do to get the polynomial function is simply apply the power rule. Let's remember that the derivative of X to the power of A is equal to a multiplied by x to the power of a minus 1. What we can do is simply divide both sides by A to isolate X on the right hand side, and we're going to get 1 divided by a multiplied by X to the power of a derivative is equal to x to the power of a minus 1. So if we simply ignore 4 and focus on X cubed, we can now show that X to the power of a minus 1 is equal to x cubed on the right hand side, right, because we have X the power of a minus 1 on the right. And 4 X cubed on the right. By ignoring 4, we can now only work with X, and we can equate our exponents to show that a minus 1 must be equal to 3. Which gives us a equals 4. So now what do we have? Well, we have shown that 1 divided by A, which is 1/4, multiplied by the derivative of X the power of 4 is equal to. A cubed. However, we have 4 x cubed within our original expression, so what we can do is simply multiply both sides by 4, which gives us 1/4 multiplied by 4. That's 1, and we are left with X to the power of 4 derivative equals 4 x cubed. We can apply the power rule to check whether or not this is correct. We bring down the exponent, which gives us 4 multiplied by X to the power of 4 minus 1, which gives us that cube. Now, this is only one of the possible antiderivatives. If we want to find the family of anti derivatives, we basically have to recall that the derivative of a constant is zero. So we can basically add a constant on the left, right? Because the derivative of a constant C is 0, and we will still get a derivative of 4 X cubed. So now combining these two derivatives, we can show that F of X is equal to. X the power of 4 + c which corresponds to the answer choice C. Thank you for watching.

Finding AntiderivativesIn Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.2x

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Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
2x