Checking Antiderivative Formulas

Verify the f...

Question

Checking Antiderivative Formulas

Verify the formulas in Exercises 57–62 by differentiation.

∫(3x + 5)⁻² dx = −(3x + 5)⁻¹/3 + C

Accepted Answer

Welcome back everyone. Determine whether the following statement is true or false. The inside derivative of 5 X + 2 to the power of -2 is 1/5 multiplied by 5 X + 2 to the power of -1 + C. A says true, and B says false. For this problem, assuming that this statement is true. We would have the derivative. Of -15 multiplied by 5 X + 2 to the power of -1 + C equal to 5 x + 2 to the power of -2. So we basically want to check if the derivative of -15 multiplied by 5 X + 2 to the power of -1 + C. is equal to, we're going to add a question mark, right? Because we're checking if this statement is true. So it would be equal to 5 X plus 2 to the power of -2. What we can do is apply the properties of derivatives. We have a sum, so we can distribute the derivative, and we can factor out our constant. So we get 1/5 multiplied by the derivative of 5 X + 2 to the power of -1. Plus the derivative of a constant c. Must be equal to 5 x + 2 to the power of -2, assuming that this statement is true. So now on the left hand side, we're going to rewrite our constant -15 and to differentiate 5X + 2 to the power of -1, we're going to apply the chain rule. First of all, let's apply the power rule using the inner function 5X + 2. We're going to get -1 multiplied by 5 X + 2, raise the power of -1 minus 1, which is -2, right? And according to the chain rule, we're multiplying by the derivative of the inner function 5 X plus 2 derivative. The derivative of a constant C is 0, so we're adding 0, right? And now let's simplify. We are going to get 1/50. Multiplied by In pri Cs, we get -1 multiplied by 5 X + 2 to the power of -2. Multiplied by 5, the derivative of 5 X is 5, the derivative of 2 is 0, right? And now simplifying, we get 1/5 multiplied by -1, which is 1/5. Multiplied by 5, this basically gives us 1, and overall we end up with 5 x + 2 to the power of -2, which is exactly what we had on the right hand side. So the answer to this problem is a true. Thank you for watching.

Checking Antiderivative FormulasVerify the formulas in Exercises 57–62 by differentiation.∫(3x + 5)⁻² dx = −(3x + 5)⁻¹/3 + C

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Checking Antiderivative Formulas

Verify the formulas in Exercises 57–62 by differentiation.

∫(3x + 5)⁻² dx = −(3x + 5)⁻¹/3 + C