10.0 Differentiation using substitution

Function

Substitution

Function

Substitution

$\sqrt {{a^2} - {x^2}} $

$x = a\sin \theta \;or\;a\cos \theta $

$\sqrt {{x^2} + {a^2}} $

$x = a\tan \theta \;or\;a\cot \theta $

$\sqrt {{x^2} - {a^2}} $

$x = a\sec \theta \;or\;a\cos ec{\kern 1pt} \theta $

$\sqrt {\frac{{a - x}}{{a + x}}} $

$x = a\cos 2\theta $

$\sqrt {\frac{{{a^2} - {x^2}}}{{{a^2} + {x^2}}}} $

${x^2} = {a^2}\cos 2\theta $

$\sqrt {ax - {x^2}} $

$x = a{\sin ^2}\theta $

$\sqrt {\frac{x}{{a + x}}} $

$x = a{\tan ^2}\theta $

$\sqrt {\frac{x}{{a - x}}} $

$x = a{\sin ^2}\theta $

$\sqrt {\left( {x - a} \right)\left( {x - b} \right)} $

$x = a{\sec ^2}\theta - b{\tan ^2}\theta $

$\sqrt {\left( {x - a} \right)\left( {b - x} \right)} $

$x = a{\cos ^2}\theta + b{\sin ^2}\theta $