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