Derivatives of sine, exponentials, logs and inverses
Test the derivative of sin x near zero, draw the tangent to ex, use ln x as the mirror image of ex to find its derivative, and carry the inverse-function rule over to arctan.
- UK: Year 12
- France: grade 11
- Iran: grade 12
- Afghanistan: grade 12
- Syria: grade 11
- Palestine: grade 12
- Ukraine: grade 11
- Russia: grade 11
Guide
(sin x)′ = cos x and (cos x)′ = −sin x. Near 0 the curve sin x hugs the line y = x, so its gradient there is 1, which is cos 0.
(ex)′ = ex: at every point the gradient of the curve equals its height. ln x is the inverse of ex, and (ln x)′ = 1x.
If f(a) = b, the inverse function has derivative (f−1)′(b) = 1f′(a) at b. The inverse curve is the mirror image of f in y = x, so the gradients are flipped.