Derivatives from tangent lines Suppose the line tangent to the...

Question

Derivatives from tangent lines Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.
b. y = f(x) / g(x)

b. y = f(x) / g(x)

Accepted Answer

Below there, today we're gonna 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 that the tangent line to the curve PF X at X equals 5 is Y equals 5 x + 2, and the tangent line to the curve of Q of X at X equals 5 is Y equals -2 x + 3. Calculate the equation of the line tangent to the curve Y equals P X divided by Q X at X equals 5. Awesome. So it appears the final answer that we're ultimately trying to solve for for this particular prompt is we're asked to calculate the equation of this line that is tangent to the curve Y equals P X divided by Q of X at X equals 5. Awesome. So with that said, we need to recall and note that in order to find the slope of the tangent line to the curve, Y equals P X. So you take say that Y equals P X divided by Q X. At and don't forget that we're trying to find, once again, we're trying to find the slope of the tangent line to the curve Y equals PFX divided by Q of X, and this is going to be at X equals 5. We need to recall and use the quotient rule. So the quotient rule, which we should recall this as P X divided by Q of X. all to prime, so we're trying to find the first derivative of P X divided by Q of X, which is going to be equal to PX multiplied by Q X minus PX multiplied by Q X all divided by Q X squared. Awesome. So thus the derivative of. Y equals P X divided by Q X X equals 5 is given as drumroll P 5 multiplied by Q 5 minus P 5 multiplied by Q. of 5, and then we need to divide everything by. P C Q 5 squared. Fantastic. So the value of the function p of X at X equals 5 will be equal to the value of the tangent line. Y equals 5 x + 2 at X equals 5. So this will be also equal to P 5 is equal to 5 of 5. Plus 2, which is equal to 27. Fantastic. So, moving right along here, the value of the function of X, so we're taking the value of Q of X now, so the value of the function Q of X at drum roll at X equals 5. It's going to be equal to the value of the tangent line, which is equal to Y. Which is equal to -2X minus 2 X plus 3, and this is going to be at X equals 5, which will also be equal to. Drum roll, Q of 5, which is going to be equal to -2 multiplied by 5. Loss 3, which is equal to -7. So, once again, when we're saying Q of 5, we're essentially saying that X is a value of 5. So we're plugging in a value of 5 in for X, just in case there's a confusion happening. So, With that said, now, our next step now is we need to take the derivative of PF X. So the derivative of PFX at X equals 5 will be equal to the slope of the tangent line, which once again our tangent line is Y is equal to 5 x + 2. So that means, without a doubt that P5 is going to be equal to 5. And the derivative of Q of X at X equals 5 is going to be equal to the slope of the tangent line, which is going to be equal to -2 X + 3. So that means that the derivative will be Q5 is going to be equal to -2. So, once again, this prime symbol indicates that it's the first derivative. Fantastic. So moving right along here. Our next step now. we need to recall a note that the value of the function, so Y equals P X divided by Q of X. At X equals 5 is going to be given as P 5 divided by Q of 5, which is going to be equal to -27 divided by 7. Fantastic. And so therefore, the derivative drum roll, so therefore the derivative of Y equals P X divided by, so when you take P X divided by Q of X is going and don't forget this is once again this is at X equals 5, is going to be given as P 5 multiplied by Q 5 minus P of 5 multiplied by Q 5, all divided by Q of 5 squared. Which will be all equal to 5 multiplied by -7 minus 27 multiplied by -2. Fantastic. And then this will be all divided by -7 square, which will be all equal to 19 divided by 49. Fantastic. So now, without a doubt. Our next step is that we need to Find the equation of the line. So now our next step is we need to find the equation of the line passing through the point. So we need to take our answer that we just, so we need to hold on to the answer that we just found, and our next step is we now need to find the equation of the line passing through the point, parenthesis 5.27 divided by 7. And we also need to note that the slope M is equal to 19 divided by 49. So once again, we're trying to find the equation of the line passing through this particular slope, sorry, passing through this particular point using our slope value. Fantastic. So, therefore, without a doubt, The equation of the tangent line to the curve. Y equals, so we're gonna say that Y equals P X divided by, so we need to take Y equals P X divided by Q X at X equals 5 is given as Y minus. So once again we're plugging in our known variables, so Y minus negative 27. Divided by 7 because once again using our XY coordinates for our coordinate point. So that's our value for Y. So now we're plugging in our known variables into the point slope form. So Y minus -27 divided by 7, and then this is gonna be equal to 19 divided by 49, multiplied by X minus 5 is our value for X is 5. So let me simplify this expression to isolate and solve for Y all by itself, using a little bit of alge we're gonna skip some steps, but your final answer should be Y is equal to -200. And 84 divided by 49. Plus 19 divided by 49 X, and that's it. That is our final answer. Hooray, we did it. So, therefore, without a doubt, our final answer once again is Y is equal to -284 divided by 49 plus 19 divided by 49 X. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Derivatives from tangent lines Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.b. y = f(x) / g(x)

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Derivatives from tangent lines Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.
b. y = f(x) / g(x)