Question

Differentiate the functions and find the slope of the tangent line at the given value of the independent variable;
f(x)=x+(9/x), x=-3

Answer 1

\[f'(x)=1-\frac{ 9 }{ x^2 }\]Plug in 3 for x to find the slope of the tangent at x = -3:\[f'(-3)=1-\frac{ 9 }{ (-3)^2 }=1-\frac{ 9 }{ 9 }=1-1=0\] Therefore, the slope of the tangent at x = -3 is 0.

@malibugranprix2000

Answer 2

@genius12 how did you figure out the derivative?

Answer 3

\[f(x) = x+\frac{ 9 }{ x }\]We know from the 'Sum Rule' for derivatives that if we are differentiating the sum of terms, then that's the same as differentiating each term separately and adding that together. So like this: If I have a function, let's say, g(x) = 2x + 3, by the Sum Rule, I will first differentiate 2x then differentiate 3 and add the two together.\[f(x)=2x+3 \rightarrow f'(x) = (2x)'+(3)'=2+0=2\]So 2x when differentiated, you are left with 2, and any constant turns in to 0, so 3 becomes 0. So the derivative is just 2.

Going back to our original equation f(x) = x + (9/x), you do the same. You differentiate x then 9/x, and the two. When you differentiate x, you just get 1. To differentiate 9/x, you use 'Quotient Rule'. It's very important to know these rules for differentiation:

Sum Rule, Difference Rule, Product Rule, Quotient Rule, Chain Rule, Power Rule, Reciprocal Rule, Constant Multiple Rule and Implicit Differentiation.

You might not know all of the above things but if you want to be able to take derivatives, then you need to know the above Rules.

Anyway, when we differentiate x + 9/x, it becomes derivative of x + derivative of 9/x. We know that derivative of x is 1 from Constant Multiple Rule or Power Rule. Derivative of 9/x is -9/x^2 from Quotient Rule.

@malibugranprix2000

Answer 4

No Medal/Fan? lol =[

Answer 5

thanks a lot @genius12