Differentiation Tool

The process of finding a derivative is called differentiation. Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. In differentiation process, we use different operators and tools for solving the problem. The operators and tools are denoted as D, ∂, and f'. (Source: Wikipedia)

General formulas and methods for differentiation:

(d / dx) (xn) = nx(n - 1)
(d / dx) (constant) = 0
(d / dx) (uv) = u (dv) / (dx) + v (du) / (dx)

Partial differentiation methods:

f (x, y) = f'x (x, y) + f'y (x, y)

Example problems for differentiation tool

Differentiation tool problem 1:

Differentiate the given function f(x) = 5x2 + 45x3 - 72x. Find the value of f'(x).

Solution:

Given function is f(x) = 5x2 + 45x3 - 72x.

Differentiate the given function with respect to x, we get

f'(x) = (5 * 2)x + (45 * 3)x2 - 72

= 10x + 135x2 - 72

Answer:

The final answer is 135x2 + 10x - 72

Differentiation tool problem 2:

Differentiate the given function f(x) = 19x4 + 35x2 - 16x. Find the value of f''(x).

Solution:

Given function is f(x) = 19x4 + 35x2 - 16x.

Differentiate the given function with respect to x, we get

f'(x) = (19 * 4)x3 + (35 * 2)x - 16

= 76x3 + 70x -16

f'(x) = 76x3 + 70x -16

For finding f''(x), differentiate the given function once again with respect to x, we get

f''(x) = (76 * 3)x2 + 70

= 228x2 + 70

Answer:

The final answer is  228x2 + 70

Differentiation tool problem 3:

Differentiate the given function f(x) = 100x5 - 42x + 102. Find the value of f'(x).

Solution:

Given function is f(x) = 100x5 - 42x + 102.

Differentiate the given function with respect to x, we get

f'(x) = (100 * 5)x4 + (42 * 1) + 0

= 500x4 + 42

Answer:

The final answer is 500x4 + 42

Practice problems for differentiation tool

Differentiation tool problem 1:

Differentiate the given function and find the f''(x). Given f(x) = 3x5 - 5x3 + 3x2 + 7

Answer:

The final answer is f''(x) = 60x3 - 30x + 6

Differentiation tool problem 2:

Differentiate the given function and find the f'''(x). Given f(x) = 16x3 - 7x2 + x

Answer:

The final answer is f'''(x) = 96