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.

−3x⁻⁴

Accepted Answer

Welcome back everyone. Find an anci derivative for the function -5 x to the power of -6. A says x to the power of 5 + c. B 6 divided by x to the power of 5 + c. C 5 divided by x the power of 5 plus c. And D 1 divided by x, the power of 5 plus c. For this problem we want to find the function F X whose derivative is lowercase f x, and it is given to us. It is -5 x to the power of -6, right? This is lowercase f x. Let's recognize that we have a constant multiplied by x raised to a constant that is not equal to -1, so we can apply the power rule. Let's remember that according to the power rule, the derivative of x to the power of a is equal to a multiplied by x to the power of a minus 1. Additionally, if we multiply both sides by some constant c, we're going to get c multiplied by x to the power of a derivative is c multiplied by a multiplied by x the power of a minus 1. So we are going to identify one of the possible antiderivatives, and then we're going to add the constant capital C to identify the whole family because the derivative of a constant is 0, right? So what we can do is simply understand that c multiplied by a multiplied by x to the power of a minus 1 is equal to -5 x to the power of -6, and immediately by analogy we can compare the exponents a minus 1. Must be equal to -6, so we can find a. A is going to be -6 + 1, which is -5. And now c multiplied by a is our constant, which is -5. So c is equal to -5 divided by a. Which is -5 divided by -5, and that's 1. So the derivative of c, which is 1 multiplied by x to the power of -5, is equal to -5 x -6. Now to get the whole family of inside derivatives, we have to add some constant c to get x to the power of -5 plus c derivative. It is also -5 x to the power of -6 because the derivative of capital C is 0, right? We can also rewrite it as 1 divided by x to the power of 5 plus c derivative is -5 x -6, meaning the correct answer is option D. 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.−3x⁻⁴

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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.
−3x⁻⁴