Differentiation Value (Math Help)

Introduction to differentiation value:

The process of finding a derivative is called differentiation. 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; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity at which the vehicle is traveling.

General formula for differentiation:

d / dx (x^n) = nx^n - 1

d / dx (constant) = 0

d / dx (Ay) = A (dy / dx)

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

d / dx (u / v) = (v (du / dx) - u (dv / dx)) / v^2

Example problems - differentiation value

Differentiation value problem 1:

Find the differentiation value of the function f(x) = 3x^5 - 5x^2 + 9x

Sol : Given function f(x) = 3x^5 - 5x^2 + 9x

Differentiate the given function with respect to x, we get

f' (x) = (3 * 5)x^4 - (2 * 5)x + 9

= 15x^4 - 10x + 9

Answer : The final answer is 15x^4 - 10x + 9

Differentiation value problem 2:

Find the differentiation value of the function f(x) = 7x^4 - 12x^3 + 12x - 76 and findf' (4).

Sol : Given function f(x) = 7x^4 - 12x^3 + 12x - 76

Differentiate the given function with respect to x, we get

f' (x) = (7 * 4)x^3 - (12 * 3)x^2 + 12 - 0

= 28x^3 - 36x^2 + 12

Substitute x = 4 in the above equation, we get

= 28 (4)^3 - 36 (4)^2 + 12

= 1228

Answer : The differentiation value is 1228

Differentiation value problem 3:

Find the differentiation value of the function f(x) = 4x^2 - 22x^3 + 56x and findf' (2).

Sol : Given function f(x) = 4x^2 - 22x^3 + 56x

Differentiate the given function with respect to x, we get

f' (x) = (4 * 2)x - (22 * 3)x^2 + 56

= 8x - 66x^2 + 56

Substitute x = 2 in the above equation, we get

= 8 (2) - 66 (2)2 + 56

= 192

Answer: The differentiation value is 192

Practice problems - differentiation value

Differentiation value problem 1:

Find the differentiation value of the function f(x) = 12x^6 - 23x^3 + 7x^2 + 89

Answer : The differentiation value is 72x^5 - 69x^2 + 14x

Differentiation value problem 2:

Find the differentiation value of the function f(x) = x^5 - 7x^4 + 12x^3 + 17x^2

Answer : The differentiation value is x^4 - 28x^3 + 36x^2 + 34x

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