Differentiation from first principles

The compound angle formulae for sin (A + B) and sin (A - B) are

sin (A + B) = sin A cos B + cos A sin B and sin (A - B) = sin A cos B - cos A sin B.

By letting C = A + B and D = A - B, show that

Hence write down a similar inequality interval for the expression

Using parts (a) and (c)(ii), show from first principles that the derivative of sin x is cos x,

where x is measured in radians.

Given that f(x) = x² + 2x, use differentiation from first principles to show that f ′(x) = 2x + 2

The gradient of a curve is given by = 2x + 2 and the curve passes through the point (- 1, 5).

Find the equation of the curve.

Conservationists are studying how the number of bees in a wildflower meadow varies according to the number of wildflower plants. The study takes place over a series of weeks in the summer. A model is suggested for the number of bees, B, and the number of wildflower plants, F, at time t weeks after the start of the study.

In the model B = 20 + 2t + cos 3t and F = 50e^0.1t.

The model assumes that B and F can be treated as continuous variables.

Suggest a reason why this model may not be valid for values of t greater than 12.

Differentiate cos x with respect to x, from first principles.

Let f(x) = 2x³ + 3x. Use differentiation from first principles to show that f ′(x) = 6x² + 3.