1. Aim: How do we calculate more complicated derivatives?

2. Simple Derivatives
Calculate the following derivates: f(x) =4x6 f(x) = 3x2 What general rule are we using?

3. Common Derivatives in the Reference Table
df/dx = df/du * du/dx d/dx (xn ) = nxn-1 d/dx (ex) = ex d/dx (lnx) = 1/x d/dx (sinx) = cosx d/dx (cosx) = -sinx

4. Graph of sine curve and its derivative

5. Graph of logarithmic function and its derivative

6. More complicated derivatives to solve:
How would we find the derivative of this function? f(x) = sin (7x)

7. Example Solved
Use the concept that df/dx = df/du * (du/dx) to find derivative of f(x) = (sin 7x)
(Let u = 7x)
If f(x) = sin7x, then f(u) = sin (u)
df/du = cos (u) and du/dx = 7
Thus, df/dx = 7 *sin(7x)

8. Solve for the following derivatives
f(x) = cos (4x) df/dx = -4sin4x
f(x) = sin (6x2) df/dx = 12xcos(6x2 )
f(x) = 13 ln (5x) df/dx = 13/x
f(x) = ln(4x3 ) df/dx = 3/x
f(x) = 25/ 1 –x df/dx = 50x/(1-x2 )2
f(x) = e4x df/dx = 4e4x
f(x) = 7esinx df/dx=7cosxesinx
f(x) = ln (11x2 ) df/dx=2/x
f(x) = cos(ex ) df/dx = -ex sin(ex )

9. Solve for the following derivatives
10) f(x) = 2x3 – 4x2 + 3x -5 df/dx=6x2-8x+30 11) f(x) = 3x + sinx – 4cos4x df/dx=3-3cosx+16sin4x 12) f(x) = x2 – 3sinx -2cosx + 4 df/dx=2x-3cosx +2sinx 13) f(x) = sin(4x) df/dx=4cos4x 15) f(x) = 3ln(x) – 4ex df/dx=3/x – 4ex 16) f(x) = 3ln(x) + x2 df/dx=3/x +2x 17) f(x) = (5+2x)2 -8 df/dx=2(5+2x)