Find the instantaneous rate of change of the following functions at x=1 and also find Stationary

step 1 In this question we are given some functions. We need to find the instantaneous rate of change o

Find the instantaneous rate of change of the following...

Key Concepts

Trigonometric Differentiation

Trigonometric differentiation involves rules for the derivatives of sine, cosine, and other trigonometric functions. For instance, the derivative of sin x is cos x and that of cos x is -sin x. These rules are crucial for analyzing periodic phenomena in fields such as physics and engineering.

Derivative

The derivative of a function at a given input measures its instantaneous rate of change, representing the slope of the tangent line to the function’s graph at that point. It is a fundamental concept in calculus that encapsulates how a function varies in response to small changes in its input, and is computed using limits of difference quotients.

Stationary Points

Stationary points, or critical points, are points on the graph of a function where its derivative is zero. These points are significant because they often correspond to local maxima, local minima, or points of inflection, and analyzing them helps in understanding the function’s behavior and optimizing problems.

Power Rule

The power rule is a basic differentiation rule for functions of the form f(x) = x^n, where n is any real number. It states that the derivative is given by f'(x) = n*x^(n-1). This rule simplifies finding the rate of change for polynomial and algebraic functions.

Chain Rule

The chain rule is a technique used to differentiate composite functions, where one function is nested inside another. It provides a systematic way of computing the derivative of a complex function by taking the derivative of the outer function and multiplying it by the derivative of the inner function.

Exponential and Logarithmic Differentiation

Differentiating exponential and logarithmic functions involves specific rules; for example, the derivative of an exponential function with base e, f(x) = e^x, is e^x, while the derivative of the natural logarithm, f(x) = ln x, is 1/x. These rules are essential when dealing with functions that exhibit rapid growth or scale changes.

Transcript

00:01 In this question we are given some functions.

00:02 We need to find the instantaneous rate of change of those functions at x equals to 1 and we need to find the stationary points.

00:10 So just look what is instantaneous rate of change? instantaneous rate of change is given by derivative of that function at that point.

00:20 Okay, so in this case we have to find the value of f -1 for all these functions.

00:25 And what is stationary points? stationary points is the point where the derivative of the function becomes 0 okay so we need to find the value of x for which the derivative of the function is becoming 0 okay so let's start from our first function our first function is given to be fx equals to x squared if we find the derivative of this function comes out to be 2x okay that means y -dash at 1 equals to 2 that means instantaneous rate of change of this function at 1 is equal to 2 and now if we find a stationary point for stationary point what we have to do is we have to put 2x equals to 0 that is y -dash equals to 0 that gives us x equals to 0 that means stationary point comes out to be 0 comma 0 because if we put the value of x is equals to 0 in the function it comes out to be 0 okay now let's look at our second function a second function is given to be x cube.

01:32 Okay, that means y -dash becomes 3x square and the value of y -dash -1 comes out to be 3.

01:39 And if we look at the stationary point, here also it comes out to be 0 -0, right? for the third function, we are given fx is equals to root x.

01:55 Okay, here y -dash becomes 1 by 2 root x, right? and y -dash -1 becomes 1 by 2 so if you look here is no stationary point because for no values of x this function can become 0 right so here we have no stationary points right now let's look on to the fourth function fourth function is given to be fx equals to 1 by x so our y -dash becomes minus 1 by x square that means y -dash 1 becomes minus 1 1.

02:32 Okay, so the instantaneous rate of change of this function at 1 is minus 1.

02:37 And here also we can see that there is no stationary points because for no values of x, y -dash can become 0.

02:45 So no stationary point.

02:50 Okay, so let's see our fifth function.

02:54 Our fifth function is given to be fx equals to e to the power x...

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