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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...

For M1, it is essential for you to be able to solve questions that involve particles moving in a straight line with constant acceleration.

The representation of symbols is listed as follow:

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f(x)=2x³+23x²+3x+5

f'(x)=6x²+46x+3

Maximum or minimum when f'(x)=0

6x²+46x+3=0

Using the Quadratic Formula: x=(-b+-squareroot(b^...

The general form for an integrand if the integral is of the form f(x^n)dx is (1/(n+1)) * x^(n+1) +c This is applied to each term in the question, remembering the constant in the integrand:

So: f(6x...