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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...

Using the definitions of coshx and sinhx (coshx=1/2(e^x+e^-x) and sinhx=1/2(e^x-e^-x)), we can substitute these into what we want to show, giving 12(1/2(e^x+e^-x)) - 4(1/2(e^x-e^x)), expanding this out gi...