Chain rule

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.

Determine the x-coordinates of any stationary points on the curve.

Show that the curve is convex for all values of x.

The diagram shows part of the curve f(x) = + 2. The equation f(x) = 0 has a positive root α close to x = 0.3.

Explain why using the sign change method with x = 0 and x = 1 will fail to locate α.

Show that the equation f(x) = 0 can be written as x =

An alternative iterative formula is x_n+1 = F(x_n), where F(x_n) = ln(2 - 8x_n²).

By considering F ′ (0.3) explain why this iterative formula will not find α.

Differentiate the following with respect to x.

(2x + 3)⁷

x³ ln x

Find the equation of the curve through (1, 3) for which = 6x - 5.

By considering show that the curve is convex for all values of x.