Differentiate following functions.

(a) f ( x ) = 15x¹⁰⁰ - 3x¹² + 5x - 46

(b) h ( x ) = x ^π  - x ^√2

 Solution

(a)    f ( x ) = 15x¹⁰⁰ - 3x¹² + 5x - 46

In this case we have the sum and difference of four terms and hence we will differentiate each term using the first property from above & then put them back together with the appropriate sign. Also, for each term with multiplicative constant remembers which all we have to do is "factor" the constant out (by using the second property) and then do the derivative.

 f ′ ( x ) = 15 (100) x⁹⁹ - 3(12) x¹¹ + 5 (1) x⁰ - 0

               = 1500 x⁹⁹ - 36 x¹¹ + 5

Notice as well that in the third term the exponent was a one & so upon subtracting 1 from the original exponent we obtain a new exponent of zero. Now recall that x⁰  = 1 .  Don't forget to do any basic arithmetic that required to be done such as any multiplication and/or division in the coefficients.

 (b) h ( x ) = x ^π  - x ^√2

         h′ ( x ) = π x ^{π -1} - √2 x  ^{√2 -1} 

The answer is a little messy and we won't decrease the exponents down to decimals.  Though, this problem is not terribly hard it just looks that way initially.

While we first put down the properties we noted that we hadn't involved a property for products and quotients. That doesn't mean that we can't differentiate product or quotient at this point. There are some which we can do.