Applications of Derivatives Class 12 Maths Solutions has been updated according to the latest syllabus of 2080. It means the notes of applications of derivatives chapter provided in this article contains all the new exercise that has recently been updated. Now you don’t need to go anywhere searching for the notes of this chapter because we are here to serve you.
Chapter – 13
Applications of Derivatives
When we talk about derivatives in this context, we often refer to the derivative of a function with respect to time or another variable, which gives us the rate at which that function is changing. For example, if $latex y = f(x)$ represents the distance traveled by an object over time $latex x$, then the derivative $latex \frac{dy}{dx}$, or $latex f'(x)$,
represents the distance traveled by an object over time $latex x$, then the derivative $latex \frac{dy}{dx}$, or $latex f'(x)$, represents the object’s velocity – how quickly the distance is changing with time. Similarly, in physics, the derivative of velocity with respect to time gives acceleration, a measure of how the velocity is changing over time.
Applications of Derivatives Class 12 Maths Solutions PDF
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Derivatives as the Rate Measure
Let $latex y = f(x)$ be the continuous function. By the definition of a function, $latex y$ changes while $latex x$ will change. If $latex \Delta x$ and $latex \Delta y$ be the small changes in $latex x$ and $latex y$ respectively, then $latex \frac{\Delta y}{\Delta x} = \frac{f(x + \Delta x) – f(x)}{\Delta x}$
is the change in $latex y$ per unit change in $latex x$ and hence is the average rate of change of $latex y$ with respect to $latex x$ on the interval $latex [x, x + \Delta x]$. The average rate of change becomes the instantaneous rate of change when $latex \Delta x \rightarrow 0$ provided that the limit exists. Thus,
$latex \lim_{{\Delta x \to 0}} \frac{\Delta y}{\Delta x} = \lim_{{\Delta x \to 0}} \frac{f(x_0 + \Delta x) – f(x_0)}{\Delta x}$. i.e,
$latex \left(\frac{dy}{dx}\right){x = x_0} = \lim{{\Delta x \to 0}} \frac{f(x_0 + \Delta x) – f(x_0)}{\Delta x}$ is the instantaneous rate of change of $latex y$ with respect to $latex x$ at $latex x = x_0$.
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