William W. answered • 12/18/20
Experienced Tutor and Retired Engineer
The value of the derivative tells you 1) the slope at a point (instantaneous rate of change) and 2) whether the function is increasing or decreasing at that point. It does not have anything to do with the function value at that point. For instance, as you mention, the slope is 6 in your function f(x) = x2 at x = 3 and f(3) = 9, we can also consider the function f(x) = x2 + 1 where the slope at x = 3 is also 6 but the function value is 10 or f(x) = x2 + 2, etc. The derivative DOES allow you to estimate a function value at a point NEAR x = 3. Given the derivative value of 6 and the point (3, 9), we can create a linear estimation of say f(3.2) by using the tangent line equation. For instance, in this case the tangent line equation is y - 9 = 6(x - 3) and we can plug in x = 3.2 to get an f(3.2) estimation: y - 9 = 6(3.2 - 3) or y - 9 = 6(.2) or y = 10.2. Of course the real value of f(3.2) is 10.24 but this process isn't bad in many cases. But it doesn't relate to the value of the function AT x = 3.
William W.
As I mentioned, you can have an entire family of functions all with the same instantaneous rate of change at a point but all shifted in the y direction from each other. One could have a function value of 1 while another has a function value of a million yet both could have an instantaneous rate of change of 6.12/18/20
Rahul A.
yeah but ultimately they both are different functions,read my first line what i want to know is the,the value ๐ is called as ๐ข๐ง๐ฌ๐ญ๐๐ง๐ญ๐๐ง๐๐จ๐ฎ๐ฌ ๐ซ๐๐ญ๐ ๐จ๐ ๐๐ก๐๐ง๐ ๐ ๐จ๐ "๐ฒ" wrt x, ๐ก๐จ๐ฐ ๐๐จ๐๐ฌ ๐ข๐ญ ๐๐ก๐๐ง๐ ๐ ๐ญ๐ก๐ ๐ซ๐๐ญ๐ ๐จ๐ ๐ญ๐ก๐ ๐๐ฎ๐ง๐๐ญ๐ข๐จ๐ง??lets talk about this function only f(x)=x^2 at x=3.12/18/20
Rahul A.
it says ๐ข๐ง๐ฌ๐ญ๐๐ง๐ญ๐๐ง๐๐จ๐ฎ๐ฌ ๐ซ๐๐ญ๐ ๐จ๐ ๐๐ก๐๐ง๐ ๐ ๐จ๐ "๐ฒ" wrt x so i thought that the slope value causes a change in y. that the value 6 is responsible for f(x)=912/19/20
William W.
The slope value does cause a change in y so you could think about the slope affecting the value of f(3.1) from f(3) (i.e., small values around x = 3) but it is the function rule that makes f(3) equal to 9.12/19/20
Rahul A.
ok thanks a lot sir. i used to think if a slope is 6 then the y value(at x=3 which here is 9) should increase by 6,but now i get the meaning its "๐ฐ๐ข๐ญ๐ก ๐ซ๐๐ฌ๐ฉ๐๐๐ญ ๐ญ๐จ ๐ฑ". i.e if x changes by 0.001, then y will change by approximately 0.006( 6 times that small amount). just one last question i might also be knowing the answer but kinda lost it, why it isnt applicable to values which are not near 3, for eg wat if x changes to 0.5,plz one last help just this query and thank you for everything.12/22/20
Rahul A.
i get it when x changes from 3 to 5 i.e x changes by 2 units then y should change 2*6=12, therefore 9+12=21 but at x=5, y=25(x^2) so there is a lot of error. as gap between points increases the error increases, now if i say x changes from 4.9 to 5 the error will be less i get 24.99 and x^2=25. my main problem was i used to think if a slope is 6 then the y value(at x=3 which here is 9) should increase by 6,but now i get the meaning its "๐ฐ๐ข๐ญ๐ก ๐ซ๐๐ฌ๐ฉ๐๐๐ญ ๐ญ๐จ ๐ฑ". "with respect to x" the meaning lies in this sentence. thanks a lot thank you very much. i hope you help me when i have queries. thank you once again.12/23/20
Rahul A.
yes thank you that was helpful,the value 6 is called as instantaneous rate of change of "y" wrt x, how does it change the rate of the function,just as i mentioned average rate change is related to y value of the function, you have said that it has nothing to do with function value , but it is called instantaneous rate of change of "y" so how is it changing the rate of the function,if it is changing then it must affect the value of the function.12/18/20