Biology Forums - Study Force

in simple terms can someone please explain the power rule and the chain rule of calculus please i have a quiz at 9am

Power rule:
the derivative of x^y is yx(y-1)

Y is any number.
Deriv. of x^2 = 2x
Deriv. of 3x^4 = 12x^3
Deriv. of 33 = 0

Dunno what the chain rule is, but I have a feeling I know it >-<

According to power rule:

If you want to differentiate x^n, i.e. d(x^n)/dx
The answer is simply n*x^(n-1)

So if you want to differentiate x^8 it equals 8*x^7

Multiply the power by x*(power-1)

Chain rule is used for functions of functions.
Let there be a function f(x) and another function g(x)
by the combination of these 2 we get f(g(x)) so to differentiate f(g(x)) we get f'(g(x))*g'(x) [f'(x) means differentiation of f(x) and g'(x) means differentiation of g(x)]
for example if we want to differentiate (sinx)^2 it is formed from 2 functions f(x)=x^2 and g(x)=sinx so f(g(x))=(sinx)^2
So the answer is this: forget the sin first. Think of (sinx) as simply x
So that means we have to differentiate x^2 which is 2x
Replace x by sinx we get 2sinx
Now multiply this by differentiation of sinx i.e. cosx


We don't actually replace x by sinx, I just did that to explain it. The point is to forget the inner function. Differentiate the outer function just like you would differentiate 'x'. And multiply the answer by the differentiation of the inner function.

in (Sinx)^2 the outer function is the square (power 2) so we differentiate that we get 2*(sinx^1)= 2*sinx (from the power rule)
we multiply this by the differentiation of the inner function (inner function is sinx differentiation of sinx is cosx)


So the answer to differentiation of (sinx)^2 is 2*sinx*cosx

Power Rule says:
derivative of x^n = n * x^(n-1)

Examples:
derivative of x^2 = 2 * x^(2-1) = 2 * x^1 = 2x

derivative of 3x^4 = 3 * 4 * x^(4-1) = 12x^3

_____________________________ _____
Suppose you have a composition of two functions, f and g, to get f(g(x)). This means you first apply function g to x, and then apply function f to g(x).

To get the derivative of f(g(x)), you use the Chain Rule.
The Chain Rule is:
derivative of f(g(x)) = f'(g(x)) * g'(x)

For example, find the derivative of 2(3x + 1)^2.
This is like 2(y)^2, where y is (3x+1). So, y' = 3.
The derivative of 2(y)^2 is 4y, so the derivative of 2(3x+1)^2 is 4(3x+1).
Then, putting it all together:
derivative of 2(3x+1)^2 = 4(3x+1) * 3 = 12(3x+1)
using the Chain Rule.

Good luck with your quiz! :)