Computing Derivatives - Derivatives of More Complicated Functions

Derivatives of More Complicated Functions

Since most functions are complicated, we need some more rules. Next: how to find derivatives of functions that
were built by taking sums, products, and quotients of simpler functions.

The "prime" notation will become more useful as the functions become more complicated. If we have some expression

□

then we can write the derivative of that expression as

(□)'.

Sample Problem

(x²)' denotes the derivative of x².

Sample Problem

If f(x) = 5x + 6, then f'(x) and (5x + 6)' mean the same thing.

Sample Problem

(4sin(x) + e^x)' denotes the derivative of 4sin(x) + e^x (we'll know how to find this soon).

Be Careful: Whenever possible, simplify the function before finding its derivative.

• Derivative of a Constant Multiple of a Function

  We'll start with taking the derivatives of some lines.

  Sample Problem

  Find the derivative of each function.

  • f(x) = x
      
  • g(x) = 3f(x)
      
  • h(x) = 3g(x)

  Answer. These are all lines.

  • The derivative of f(x) = x is the slope of the line f(x) = x, which is 1. Therefore f ' (x) = 1.
      
  • Since f(x) = x,
      
    g(x) = 3f(x) = 3x.
      
    This is a line with slope 3, therefore g ' (x) = 3.
      
  • Since g(x) = 3x,
      
    h(x) = 3g(x) = 3(3x) = 9x. 
    This is a line with slope 9, therefore h ' (x) = 9.
      

  If we multiply a function by 3, the derivative gets multiplied by 3 also. If we multiply a function by 2, the derivative gets multiplied by 2. And so on. The derivative of the function cf(x), assuming f is differentiable, is cf ' (x).

  In symbols, if g(x) = cf(x) where c is a constant, then 

  g ' (x) =cf ' (x).

  In words, if we have a function f and multiply it by some constant c to find a new function, then the derivative of that new function is c multiplied by the derivative of f. This works for any differentiable function f and any constant c.

  In pictures, it's easiest to see what's going on with a line. If we take a line y = mx + b, it looks something like this:

  

  If we multiply the whole line by 3, the line gets stretched vertically:

  

  Now the line is 3 times steeper. Although it's a bit harder to see the picture with curvy functions, the idea is the same. If we stretch (or shrink) the function vertically, we're also stretching (or shrinking) its derivative.

  

• Multiplication by -1

  Remember that putting a negative sign in front of a function means the same thing as multiplying that function by -1.

  Sample Question

  Let g(x) = -x². Then we could think of this function as

  g(x) = (-1)(x²),

  therefore

  g ' (x) = (-1)(x²)' = (-1)(2x) = -2x.

  The moral of the story is that the derivative of the negative of f is the negative of the derivative of f:

  (-f(x))' = -f ' (x).

  Sample Problem

  Let f(x) = -sin x. Then 

  f ' (x) = -(sin x)' = -cos(x).

  This is really just a special case of the constant multiple rule, given that -1 is a constant and all.

• Fractions With a Constant Denominator

  If we have a fraction  where c is a constant, this is the same thing as . When asked to take derivatives of functions like this, the first thing we do is rewrite the original function to make it easier to see what we should do with it.

  We can do the same thing when we have other less complicated functions in the numerator.

• More Derivatives of Logarithms

  Whenever possible, rewrite a function so it looks good before taking its derivative.

  Any logarithm log_b a can be written as

  

  This allows us to find derivatives of logarithms in bases besides e.

  We also know enough now to find derivatives of things like

  f(x) = ln (x²).

  We can bring down the exponent to find

  f(x) = 2 ln x,

  and we know how to find the derivative of this: