Derivatives and tangent lines
a. For the following funct...

Question

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = √3x; a= 12

Accepted Answer

Hey everyone, and welcome back to another video. Calculate the derivative of the function F of X is equal to square root of 5 X X equals 25. We're given 4 answer choices a square root of 5 divided by 5, B square root of 5 divided by 15, C square root of 5 divided by 10, and the square root of 5 divided by 2. So first of all, let's see that F of X is equal to square root of 5 X. And because we want to find its derivative, we can factor out the constant because square root of 5 X is simply square root of 5 multiplied by. square root of X and this can be rewritten as square root of 5 multiplied by x the power of 1/2 because we simply want to apply the power rule to differentiate X the power of 1/2, right? So if we find the derivative, we can factor out the constant which is square root of 5, and then we are looking for the derivative of X to the power of 1/2. So first of all, we get square root of 5, which is our constant, and now applying the power rule, we have to bring down the exponent, which is 1/2, and it becomes our coefficient, and then we're multiplying by X the power of 1/2 minus 1 based on the rule, which gives us 1/2 as the new exponent. And now simplifying, we can say that this is square root of 5 divided by. 2 because we have 2 in the denominator, and now we also get X because it has a negative exponent so we can turn it into a positive exponent in the denominator and now we can turn it back into square root. So we get square root of 5 divided by 2 square root of X. And we can also rationalize it. Let's multiply both sides by square root of X, which gives us square root of 5 X divided by 2 X. So from here, what we can do is evaluate the derivative at X equals 25. We want to evaluate F at 25. So for every X we simply want to substitute 25. In the numerator, we get 5 multiplied by 25, and in the denominator, we get 2 multiplied by 25. So, the numerator gives us square root of 5 squared multiplied by 55 squad is 25, right? And then the denominator, well, we get 50. So now let's simplify. We can take square root of 5 squad, which is 5, and we're left with square root of 5, so that's our numerator and in the denominator, we have 150. So now let's divide both sides by 5, which gives us 1 in the numerator and some in the denominator, which means that the final answer is square root of 5 divided by sun. And this corresponds to the answer choice C. Thank you for watching.

Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = √3x; a= 12

Key Concepts

Derivatives

Tangent Lines

Function Evaluation

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