Simple derivatives

Example   Differentiate following.

 (5x³  - 7 x + 1)⁵ ,[ f ( x )]⁵ ,[ y ( x )]⁵

Solution: Here , with the first function we're being asked to do the given,

d  [(5x³  - 7 x + 1)⁵/ dx =  5 (5x3  - 7 x + 1)⁴ (15x²  - 7 )

and it is just the chain rule.  We differentiated the outside function (the exponent of 5) and then multiplied that by the derivative of the inside function (the stuff inside the parenthesis).

For the second function we're going to do basically the similar thing.  We're going to have to use the chain rule. Still the outside function is the exponent of 5 whereas the inside function this time is simply f ( x ) . We don't contain a particular function here, however that doesn't mean that we can't at least write down the chain rule for this function. Following is the derivative for this function,

d  [ f ( x )]5 / dx = 5 [ f ( x )]⁴ f ′ ( x )

Actually we don't know what f ( x ) is so while we do the derivative of the inside function all we can do is write down notation for the derivative, that means f ′ ( x ) .

Along with the final function here simply we replaced the f in the second function along with a y since most of our work in this section will involve y's rather than f's.  Outside of that this function is alike to the second. Thus, the derivative is,

d  [ y ( x )]5  = 5 [ y ( x )]⁴  y′ ( x )