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.

-(1/2)x⁻³ᐟ²

Accepted Answer

Welcome back everyone. Find the side derivative of 4 x the power of -3 halves. A 8 x, the power of 1/2 plus c. B 8 x, the power of -1/22 plus c. C 1/8 x, the power of 1/2 plus c. And D x, the power of -1/2 plus c. For this problem we once find the anci derivative capital F of X, whose derivative is 4 x to the power of -3 halves, and what we can do is simply remember the power rule. The derivative of c multiplied by x, the power of a is equal to c multiplied by a multiplied by x, the power of a minus 1. And in this case, looking at the right hand side, we know that CA x, the power of a minus 1 is equal to 4 x, the power of -312. Which allows us to identify the exponent immediately since we have a minus 1 is equal to -3 halves, right? So by analogy we can identify the value of a. Eating a minus 1 and -3 halves. We get A equals -3 halves plus 1, which is equal to -1/22. And now considering the constant c, we can equate c a and 4 because this is our coefficient, right? So if c multiplied by a is equal to 4, c is equal to 4 divided by a, which is 4 divided by 1/22. Which is equal to -8. So now we have identified C, which is -8, and A, which is 1/22. Meaning the derivative of 8 x to the power of 1/22. Is equal to 4 x the power of -3 halves. From here let's remember that the derivative of a constant is zero. So to find the whole family of anti derivatives we can basically add a constant to show that the derivative of 8 x to the power of -1/2 plus some constant c, the derivative of any function in this form for any constant capital C, right. We will have A derivative of 4 x, the power of -3 halves as well. So the correct answer for this problem corresponds to the answer choice b. 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.-(1/2)x⁻³ᐟ²

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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.
-(1/2)x⁻³ᐟ²