Question

Hello,

Could somebody please help me with the following question? It asks to differentiate the function below according to derivate rules of calculus such as the power rule (if f(x)=x^n, then f'(x)=nx^n-1), the product rule (if F(x)=f(x)g(x) then F'(x)=f(x)g'(x)+f'(x)g(x)) and the chain rule for polynomials (if F(x)=(f(x))^n then F'(x)=nf'(x)f(x)^n-1).

Here's the function:

f(x)=3x/x^2+4

My tentative solution:
f(x)=3x/x^2+4
f(x)=(3x)(x^2+4)^-2
f'(x)=(3x)(-2)(x^2+4)(x^2+4)^-2-1 (chain rule)
f'(x)=(3x)(-2)(x^2+4)(x^2+4)^-3
f'(x)=(-6x)(x^2+4)(x^2+4)^-3

I'm not sure whether to apply the chain rule of the product rule in step 2 or both of them.

Any help would be much appreciated!

Constantin

Answer 1

you need both the product rule and the chain rule.

f(x) = (3x)(x^2+4)^-1
f'(x) = (3)(x^2+4)^-1 + (3x)(-1)(x^2+4)^-2(2x)
= 3/(x^2+4) - 6x^2/(x^2+4)^2
= (3(x^2+4)-6x^2)/(x^2+4)^2
= (12-3x^2)/(x^2+4)^2

The quotient rule produces the same result; in fact, you can see the intermediate steps in the calculations above, if you look carefully.

Answer 2

Thank you Steve! I understand it now.

Answer 3

Given the function rule f(x) = 2x ^ 2 - 7x + 1 what is the

output of f(- 3) :

Answer 4

Thanks u a life saver

Answer 5

To find the output of f(-3), we need to plug in -3 for x in the function rule f(x):

f(-3) = 2(-3)^2 - 7(-3) + 1

Simplifying the expression:

f(-3) = 18 + 21 + 1

f(-3) = 40

Therefore, the output of f(-3) is 40.

Answer 6

You're welcome! I'm glad I could help.