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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...

this formula is used in triangles to define either a cosine or the side of the triangle opposite of the cosine's angle

c²=a²+b²-2abcosc

cosc= the cosine of t...

Gradient of initial line = 3/4 Equation of initial line : y= 3/4x + 5/4 (4y = 3x + 5) Equation of perpendicular line : y= -4/3x 

a.) simplify and expand (x+3)(2x+5) Expand using "crab claw" (better known as foil) => 2x^2 + 5x + 6x + 15 Simplify => 2x^2 +11x + 15 b.) differentiate (x+3)(2x+5) Using part a,  dy/dx= 4x...