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) = 1/3x-1; a= 2

Accepted Answer

Welcome back, everyone. Calculate F at A4 F of X equals 1 divided by 4 X minus 51 A is equal to 3. We're given 4 answer choices A says -4 divided by 49, B 4 divided by 49, C-4 divided by 7, and D 4 divided by 7. So let's begin by solving for F. We want to find the first derivative of F, and we're going to say F of X. So we want to identify the derivative of 1 divided by 4 X minus 5. And we have to recognize that this is a composite function because we can rewrite it as 4 X minus 5 raise to the power of negative first. Now we have an exponential function, right, and then we have a linear function inside. So if we want to find a derivative of a composite function, well, simply speaking, what we want to do is apply the chain rule, and we're going to set u equals 4 x minus 5 our inner function. So the derivative of F, let's call it D Y. Divided by the X is going to be found by differentiating the function with respect to you. And then multiplying the derivative of you with respect to x. This is the chain rule. And what we're going to do is identify the derivative of Y with respect to you. Notice that our function F of you now becomes U to the power of negative first, right? So here we need to apply the power rule, right? So we're going to say that this is equal to. -1, we're bringing down the exponent. And then our original exponent decreases by 1 unit, so -1 minus 1 gives us -2. And then we're multiplying by the derivative of U with respect to X. So we want to identify the derivative of 4 X minus 5, right? So let's say that we're evaluating the derivative of 4 X minus 5. And now let's substitute you back in so we get negative. 4 X minus 5 to the power of negative multiplied by 4. Because the derivative of 4 x minus 5 is 4, the derivative of a constant is 0, we factor out the constant in the derivative of X is 1. And now simplifying, we get negative or divided by because we have a negative exponent. 4 x minus 5 squared in the denominator, so we turn that negative exponent into a positive one. And now we want to identify F at A, so F at. 3. So we take the derivative that we have found and substitute X equals 3, so we get -4 divided by 4, multiplied by 3. Minus 5 squared. All that we have to do from here is simplify the result. -4, this is what we have in the numerator. In the denominator, we get 12 minus 5, which is 77 squad is 49. So -4 divided by 49 is our final answer, which corresponds to the answer choice A. Thank you for watching.

Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = 1/3x-1; a= 2

Key Concepts

Derivatives

Tangent Lines

Function Evaluation

Watch next