Relationship between First and Second Derivatives of a Function
Key Questions

Since
#f''# is the first derivative of#f'# ,#f''# tells us about the increasing/decreasing behavior of#f'# . 
Answer:
#2sec^2xtanx# Explanation:
First we find
#d/dxtanx# .We know that
#tanx=sinx/cosx# So we can use the quotient rule to solve for this:
#d/dx(sinx/cosx)=(cosxd/dx(sinx)sinxd/dx(cosx))/cos^2x# #color(white)(d/dx(sinx/cosx))=(cosx(cosx)sinx(sinx))/cos^2x# #color(white)(d/dx(sinx/cosx))=(cos^2x+sin^2x)/(cos^2x)=1/cos^2x# #d/dx(sinx/cosx)=d/dxtanx=sec^2x# Now for
#d^2/dx^2tanx# , or#d/dxsec^2x# Which we can write as
#d/dx(secx)^2# , which gives:#2secx(secxtanx)# , using the chain rule, where we compute#d/(du)u^2# and#d/dxsecx# .Which gives:
#2sec^2xtanx# So:
#d^2/dx^2tanx=2sec^2xtanx# 
Answer:
See below.
Explanation:
The second derivative is the derivative of the derivative of a function.
Let's take a random function, say
#f(x)=x^3# . The derivative of#f(x)# , that is,#f'(x)# , is equal to#3x^2# .The second derivative of
#x^3# is the derivative of#3x^2# . That's#6x# .So we say that the second derivative of
#f(x)=x^3# , or#f''(x)# , is equal to#6x#
Questions
Graphing with the Second Derivative

Relationship between First and Second Derivatives of a Function

Analyzing Concavity of a Function

Notation for the Second Derivative

Determining Points of Inflection for a Function

First Derivative Test vs Second Derivative Test for Local Extrema

The special case of x⁴

Critical Points of Inflection

Application of the Second Derivative (Acceleration)

Examples of Curve Sketching