Find the first and second derivatives of the functions in Exer...

Question

Find the first and second derivatives of the functions in Exercises 33–38.

s = (t² + 5t − 1) / t²

Accepted Answer

Welcome back, everyone. Determine the 1st and 2nd derivatives F of X equals 3 x2 minus 4 X plus 2 divided by X2. Before we begin, let's simplify the function by performing division. 3 x 2 divided by X2 gives us 3. We're then subtracting 4 X divided by X2, which is 4 divided by X, and then we are adding 2 divided by X2. We can rewrite this as 3 minus 4 X raises the power of -1 using the reciprocal rule plus 2 multiplied by x to the power of -2 using the same reciprocal rule. And then we can identify the first derivative using the power rule. So we are differentiating 3. Subtracting 4 multiplied by the derivative of X the power of -1. And adding to multiplied by the derivative of X raises the power of -2. The derivative of 3 is 0 because it's a constant. We are then subtracting 4 multiplied by. Applying the power rule, we bring down the exponent, which is -1. In the original exponent of X decreases by 1 unit, so we get X to the power of -2. We apply the same rule for the next term. We get 2 as our constant multiplied by -2, X raises to the power of -2 minus 1, which is -3. And now we can rewrite this as or X the power of -2 minus 4. X raised the power of -3. Using the reciprocal rule, we can also say that this is equal to 4 divided by X2 minus 4 divided by X cubed. Using the common denominator, we can see that it is equal to X cubed. So the first term becomes 4 X. And then we are subtracting 4, and now we can also factor it out. That'd be 4 imperies X minus 1 divided by X cubed. That would be our first derivative. Let's label it as our final answer, and now let's continue. And finds the second derivative by differentiating the answer from the first part. And what we're going to do is simply apply the power rule again. We have 4 multiplied by the derivative of X, the power of -2. -4 multiplied by the derivative of X raises to the power of -3. Applying the power rule, we get 4 multiplied by -2 X, raise the power of -3, minus 4 multiplied by -3, we bring down the exponent. Xs erase the power of -4. Now let's simplify. We get -8 X to the power of -3 plus 12 X raise to the power of -4. Let's apply the reciprocal rule. This gives us -8 divided by X cubed. Plus 12 divided by X to the power of 4, and now finding the common denominator, we can see that it is X to the power of 4. And then we simply have 12 minus 8 X in the numerator. We can also factor it out. Specifically, we can seek out 4. And er we're going to have 3 minus 2 X. Divided by x to the power of 4, which is our second derivative. Well then we have our final answers and thank you for watching.

Find the first and second derivatives of the functions in Exercises 33–38.

s = (t² + 5t − 1) / t²

Key Concepts

Derivatives

Quotient Rule

Simplifying Functions

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