Finding Derivative Functions and Values

Question

Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.

p(θ) = √3θ; p′(1), p′(3), p′(2/3)

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Given the function R of theta is equal to the square root of 6 theta. Find R of theta using the limit definition of the derivative, then calculate the value of our of 2. Awesome. So it appears for this particular problem we're asked to solve for two separate answers. We're asked to find our prima theta, that's our first answer we're trying to solve for using the limit definition of the derivative. And our second answer is we're asked to calculate the value of our of 2. So now that we know that we're trying to solve for two separate answers, our first step to solve this particular problem is we need to recall and use the limit definition of the derivative, which, as we should recall, this states that R of theta for this particular case, so our prime of theta is equal to the limit as H approaches 0 of R. Of theta plus H minus R of theta, all divided by H. Fantastic. So now our first step towards solving this particular problem is we need to substitute the given function into our limit definition. So we need to recall and note that R of theta. is equal to the square root of 6 theta, and we also need to note that R of theta plus H is going to be equal to the square root of 6 multiplied by theta plus H. So therefore, we could go ahead and write that R of theta is going to be equal to the limit as H approaches 0 of the square root of 6 multiplied by theta plus H minus the square root of 6 theta divided by H. So, Therefore, without a doubt. That is how that's we can also simplify it. So therefore we could go ahead and write that R theta is equal to the limit, as H approaches 0 of drum rule, the square root of 6 theta plus 6 H minus the square root of 6 theta divided by H when we simplify. So now our third step is we need to recall and use the conjugate method, so meaning we need to multiply both the numerator and the denominator by the square root of 6 theta plus 6 H plus the square root of 6 theta. And when we do that, we will get R. Of theta is equal to the limit as H approaches 0 of the square root of 6 theta plus 6 H. And then we need to take the square root of 6 theta plus 6 H, and we need to multiply it by parentheses, the square root of 6 theta plus 6 H plus the square root of 6 theta. Fantastic. And then we need to divide everything by H, multiplied by parentheses, the square root of 6 theta plus 6 H, and then we need to multiply, actually we're gonna add, so we're gonna take H multiplied parentheses by the square root of 6 theta plus 6 H plus the square root of 6 theta and then don't forget your parenthesis. Awesome. So now, as we should recall, we need to recall and use the difference of squares formula, which states that A minus B in parentheses multiplied by A plus B in parentheses is equal to A2 minus B2. So, using our difference of square formula, we go ahead and write that R of theta is equal to the limit as H approaches 06 theta plus 6 H minus 6 theta, all divided by H multiplied by the square root of 6 theta plus 6 H. Plus, the square root of 6 theta, fantastic. And when we do a little bit of simplifying, we will find that this is equal to the limit as H approaches 0 of drum roll. 6 H divided by H multiplied by the square root of 6 theta plus 6 H. Plus the square root of 6 theta. Fantastic. So moving right along here. So now, our 4th step is we need to cancel out H. So, if we cancel H in the numerator and the denominator, we will find that our prime of theta is going to be equal to the limit, as H approaches 0 of 6 divided by the square root of 6 theta plus 6 H plus the square root of 6 theta, fantastic. So now, our 5th step is we need to take the limit as H approaches 0, so we need to note that as H approaches 0, that will mean that as a result, the square root of 6 theta plus 6H is going to approach drum roll, the square root of 6 theta. Fantastic. So, therefore, as a result, we could go ahead and write that R of theta is going to be equal to 6 divided by the square root of 6 theta plus the square root of 6 theta. So when we simplify this expression down to its most simple form, which I'm going to skip quite a few steps, because you just need to recall and use algebraic methods to simplify our expression of 6 divided by the square root of 6 theta plus the square root of 6 theta, and you write it in its most simple form, you will get the square root of 6 theta divided by 2 theta. Fantastic. So now, Our next step So our next step now is we need to drum roll. We need to note that we need to calculate our prime. Of 2. So once again, we need to quickly recap. So we found our first answer. We found R theta, which is the square root of 6 theta divided by 2 theta. So now our next step is we need to calculate R of 2. So that means, let's make a note here, so that means we need to substitute in a value of theta equals. 2 into our expression, so plugging in a value of 2 in for theta, we will get that R of 2. It's going to be equal to the square root of 6 multiplied by 2 divided by 2 multiplied by 2. And when we plug that into a calculator and we write it in its simple form, we will get the square root of 3 divided by 2, and that will be our final answer. Hooray, we did it. So, therefore, we found our answer for our prime of theta, and we also found our answer for our prime of 2. Hooray, we did it. So let's go back up to the top to look at our problem itself, to recap everything that we've gone over to see what we solve for officially. So that will mean that our final two answers, without a doubt, will have to be R of theta is equal to the square root of 6 theta divided by 2 theta and R of 2 is equal to the square root of 3 divided by 2. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Finding Derivative Functions and Values

Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.

p(θ) = √3θ; p′(1), p′(3), p′(2/3)

Key Concepts

Definition of Derivative

Power Rule for Derivatives

Substitution in Derivatives

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