Relationship between First and Second Derivatives of a Function

Key Questions

What is the relationship between the First and Second Derivatives of a Function?

Since $f''$ is the first derivative of $f'$ , $f''$ tells us about the increasing/decreasing behavior of $f'$ .

Wataru · · Aug 30 2014
How do you find the second derivative of $y=tan(x)$ ?

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:

$d^2/dx^2tanx=2sec^2xtanx$

Surya K. · 5 · Mar 4 2018
What is a second derivative?

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 $6x$

Surya K. · 1 · Mar 10 2018