Hello,

Question

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

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 5

Thanks u a life saver

Answer 6

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