Find the derivatives of the functions in Exercises 1–42.

Question

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s = √ t .

1 + √ t

Accepted Answer

Welcome back, everyone. Calculate the derivative of the function F of X equals square root of x divided by 5 minus square root of X. We want to identify F of X, and we can do that using the quotient rule because we have a rational function. The quotient rule says that first of all, we are taking the derivative of the top function. So we're going to take the derivative of square root of X, multiply it by the bottom function, which is 5 minus square root of X. We're going to subtract the top function square root of X, which is multiplied by the derivative of the bottom function 5 minus square root of X, and all of that has to be divided by the square of the bottom function, so 5 minus square root of X squared. Now what we have to do is simply recall that the derivative of square root of X is 1 divided by 2 square roots of X. We can arrive at this result from the power rule because X to the power of 1/2 is equal to square root of X, and applying the power rule we're going to get 1/2 x raises the power of -12, which is equal to 1 divided by 2 square root of X. Those expressions are equivalent. So now we can conclude that F of X is going to be equal to. The derivative of square root of X. Which is 1 divided by 2 square root of X, multiplied by 5 minus square root of X, minus square root of X multiplied by the derivative of 5 is 0. We are also subtracting a negative term. So we're going to add a positive sign, and then let's add the derivative of square root of X, which is 1 divided by 2 square roots of X. And now considering the denominator, we have 5 minus square root of X2. All that we have to do is simplify. So what we can notice is that in the numerator, square root of X is divided by square root of X, those terms can be canceled out, right? And essentially our sum is going to have a denominator of 2 and also square root of X, right? So what we want to do is identify the common denominator. Let's write our fraction sign. We're going to have 5 minus square root of X. Divided by 2 square root of X, this is going to be our first term written as a single fraction, and then we are adding 1 divided by 2 or 12, right? So essentially we originally have 1/2, and if we want to identify the common denominator, we simply want to multiply and divide both sides by square root of X. So we're going to add square root of X for each side. And in the denominator we still have 5 minus square root of X squared. Now let's combine the like terms, 5 minus X plus of X is just 5. And because we have a double division, we can just combine those denominators in a single denominator, so we're going to have 2 square root of X multiplied by 5 minus square root of X squad, which is our final answer to this problem. So let's label it and thank you for watching.

Find the derivatives of the functions in Exercises 1–42.
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s = √ t .
1 + √ t

Key Concepts

Derivative

Chain Rule

Power Rule

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