Step by Step Differentiation

In calculus (a branch of mathematics) the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; the process of finding a derivative is called differentiation. The reverse process is called antidifferentiation (Source: Wikipedia)

General formula for differentiation:

`(d / dx)` (xn) = nx(n - 1)

`(d / dx)` (uv) = u `((dv) / (dx))` + v `((du) / (dx))`

Example problems for step by step differentiation

Step by step differentiation example problem 1:

Differentiate the given function u = 4x3 + 3x2 + 245x. Find the second derivative value of the given function.

Solution:

Given function is u = 4x3 + 3x2 + 245x

Step 1:

Differentiate the given function u with respect to x, we get

`((du) / (dx))` = (4 * 3)x2 + (3 * 2)x + 245

= 12x2 + 6x + 245

Step 2:

Differentiate the above value `((du) / (dx))` with respect to x, we get the second derivative value

`((d^2u) / (dx^2))` = (12 * 2)x + 6

= 24x + 6

The second derivative value of the given function is 24x+ 6

Answer:

The final answer is 24x + 6

Step by step differentiation example problem 2:

Differentiate the given function v = 9x2 + 12x. Find the second derivative value of the given function.

Solution:

Given function is v = 9x2 + 12x

Step 1:

Differentiate the given function u with respect to x, we get

`((dv) / (dx))` = (9 * 2)x + 12 + 0

= 18x + 12

Step 2:

Differentiate the above value `((dv) / (dx))` with respect to x, we get the second derivative value

`((d^2v) / (dx^2))` = 18 + 0

= 18

The second derivative value of the given function is 18

Answer:

The final answer is 18

Step by step differentiation example problem 3:

Differentiate the given function v = 11x4 + 41x3. Find the third derivative value of the given function.

Solution:

Given function is v = 11x4 + 41x3

Step 1:

Differentiate the given function u with respect to x, we get

`((dv) / (dx))` = (11 * 4)x3 + (41 * 3)x2

= 44x3 + 123x2

Step 2:

Differentiate the above value ((dv) / (dx)) with respect to x, we get the second derivative value

`((d^2v) / (dx^2))` = (44 * 3)x2 + (123 * 2)x

= 132x2 + 246x

Step 3:

Differentiate the above value with respect to x, we get the third derivative value

`((d^3v) / (dx^3))` = (132 * 2)x + 246

= 264x + 246

The third derivative value of the given function is 264x + 246

Answer:

The final answer is 264x + 246

Practice problems for step by step differentiation

Step by step differentiation practice problem 1:

Find the first derivative of the given function f (x) = 10x2 - 782x

Answer:

The final answer is f' (x) = 20x - 782

Step by step differentiation practice problem 2:

Find the Second derivative of the given function f (x) = 1.7x3 + 82x2 + 37

Answer:

The final answer is f'' (x) = 10.2x + 164