The derivative, events the price at which the needy variable changes with revere to the independent variable. It is single of the most significant ideas in Calculus. The quality differentiations of functions are extensively used in science, finances, medicine and computer science. Additional split the class of continuous functions into two sub classes, derivable and non-derivable. After having continuous functions, limits and stability, we shall further split the group of continuous functions into two sub classes, derivable and non-derivable.
Formula for Quality Differentiation
f'(x) = Lim f(x+∂x) – f(x)
∂x → 0 ——————–
∂x
This equation is well known as quality differentiation equation. By use this equation we preserve get derivatives of dissimilar functions. ∂x is the little modify in the variable x. f(x) is a purpose of x. f ‘(x) is derivative of f(x).
Example Problems for Quality Differentiation
Ex: 1 Find the derivative of 8x + 10
Sol:
Let f(x) = 8x + 10 then f(x + ∂x) = 8(x + ∂x) + 10
f'(x) = Lim f(x+∂x) – f(x)
∂x → 0 ——————–
∂x
= Lim 8(x + ∂x) + 10 – (8x + 10)
∂x → 0 —————————
∂x
= Lim 8x + 8∂x + 10 – 8x -10
∂x → 0 ————————
∂x
= Lim 8
∂x → 0
= 8.
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Ex 2: Find the derivative of 5x + 20
Sol:
Let f(x) = 5x + 20 then f(x + ∂x) = 5(x + ∂x) + 20
f'(x) = Lim f(x+∂x) – f(x)
∂x → 0 ——————–
∂x
= Lim 5(x + ∂x) + 20 – (5x + 20)
∂x → 0 —————————
∂x
= Lim 5x + 5∂x + 20 – 5x -20
∂x → 0 ————————
∂x
= Lim 5
∂x → 0
= 5.
Ex 3: Find the derivative of the equations f(x) = 3 + 5cos x, find f ‘(x).
Sol:
f'(x) = d(3 + 5cos x)
————–
dx
= d(3) + d(5cos x)
—- ———
dx dx
= 0 + 5(-sin x)
= -5sin x.
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