Search over 10,000 free study notes

We will be using the quotient rule, although the product rule is also usable and can be run through if the student wishes. Firstly, define u = 8x, v = x-8 for simplicity. Then clearly u' = 8, v' = 1, and ...

For this we must use the chain rule. We start by defining x³ as a new variable, u = x³ Can then rewrite the expression as y = sin(u) Chain rule tells us that dy/dx = (dy/du)(du/dx) W...

As both equations are equal to y, we can combine them to create a single equation in terms of x: x^3 - x^2 -5X + 7 = x + 7. Shift the equation so the left hand side is equal to 0 on the right: x^3 - x^2 -...

First, differentiate and put the derivative equal to zero. dy/dx=6x^2-30x+24=0. Solve this equation to get that x=4 and x=1. Substitute these values into the original equation to get the correspo...

From the definition of a derivative: f'(x) = lim h->0 ((f(x+h) - f(x)) / h) Let f(x) = x^n --> d\dx x^n = lim h->0 (((x+h)^n - x^n) / h) By binomial expansion, (x+h)^n = x^n + nhx^(n-1) + n(n-1)h...