# differential calculus examples

We will start in this section with some of the basic properties and formulas. In other words, we can “factor” a multiplicative constant out of a derivative if we need to. There are some that we can do. That doesn’t mean that we can’t differentiate any product or quotient at this point. If f is a differentiable function on ℝ (or an open interval) and x is a local maximum or a local minimum of f, then the derivative of f at x is zero. For example, velocity is the rate of change of distance with respect to time in a particular direction. Limits are used to define the continuity, integrals, and derivatives in the calculus. It is an easy mistake to “go the other way” when subtracting one off from a negative exponent and get $$- 6{t^{ - 5}}$$ instead of the correct $$- 6{t^{ - 7}}$$. Integral Calculus joins (integrates) the small pieces together to find how much there is. It will be tempting in some later sections to misuse the Power Rule when we run in some functions where the exponent isn’t a number and/or the base isn’t a variable. Next, let’s take a quick look at a couple of basic “computation” formulas that will allow us to actually compute some derivatives. It is still possible to do this derivative however. Now, we can see that these two points divide the number line into three distinct regions. This is not something we’ve done to this point and is only being done here to help with the evaluation in the next step. When you see radicals you should always first convert the radical to a fractional exponent and then simplify exponents as much as possible. Now recall that $${x^0} = 1$$. The third proof is for the general rule but does suppose that you’ve read most of this chapter. We need the derivative in order to get the velocity of the object. Once we know this we also can answer the question. In each of these regions we know that the derivative will be the same sign. Example 2: f(x) = x 3. Following this rule will save you a lot of grief in the future. Applications of derivatives. Back when we first put down the properties we noted that we hadn’t included a property for products and quotients. Here is the function written in “proper” form. Let’s graph these points on a number line. g(z) = 4z7 −3z−7 +9z g ( z) = 4 z 7 − 3 z − 7 + 9 z Solution. Make your first steps in this vast and rich world with some of the most basic differentiation rules, including the Power rule. The limit is an important thing in calculus. In Maths, when one or more functions and their derivatives are related with each other to form an equation, then it is said to be a differential equation. Differential Calculus Basics. Take an example, if f(x) = 3x be a function, the domain values or the input values are {1, 2, 3} then the range of a function is given as. It is still possible to do this derivative however. The method that we tend to prefer is the following. Since polynomials are continuous we know from the Intermediate Value Theorem that if the polynomial ever changes sign then it must have first gone through zero. Here is the function written in “proper” form. So, upon evaluating the derivative we get. For problems 1 – 12 find the derivative of the given function. Differential calculus arises from the study of the limit of a quotient. If f(x) is a function, then f'(x) = dy/dx is the. If you need some review or want to practice these kinds of problems you should check out the Solving Inequalities section of the Algebra/Trig Review. From x2+ y2= 144 it follows that x dx dt +y dy dt = 0. h(y) = y−4−9y−3 +8y−2 +12 h ( y) = y − 4 − 9 y − 3 + 8 y − 2 + 12 Solution. Let’s compute some derivatives using these properties. We will give the properties and formulas in this section in both “prime” notation and “fraction” notation. In each of these regions we know that the derivative will be the same sign. The derivative is used to show the rate of change. Here is the derivative. It is one of the major calculus concepts apart from integrals. Also, don’t forget to move the term in the denominator of the third term up to the numerator. That is usually what we’ll see in this class. In a later section we will learn of a technique that would allow us to differentiate this term without combining exponents, however it will take significantly more work to do. All of the terms in this function have roots in them. The derivative, and hence the velocity, is. Luckily for us we won’t have to use the definition terribly often. Important Questions Class 12 Maths Chapter 9 Differential Equations. Suppose we have a function f(x), the rate of change of a function with respect to x at a certain point ‘o’ lying in its domain can be written as; So, if y = f(x) is a quantity, then the rate of change of y with respect to x is such that, f'(x) is the derivative of the function f(x). Or you can consider it as a study of rates of change of quantities. Thus when x(t) = 4 we have that y(t) = 8 p 2 and 4 1 2 +8 2 dy dt = 0. It will be tempting in some later sections to misuse the Power Rule when we run in some functions where the exponent isn’t a number and/or the base isn’t a variable. As we saw in those examples there was a fair amount of work involved in computing the limits and the functions that we worked with were not terribly complicated. These are the only properties and formulas that we’ll give in this section. Luckily for us we won’t have to use the definition terribly often. Therefore, the range of the function will be {3, 6, 9}. 5 0 obj We can now differentiate the function. Differentiation has many applications in various fields. Before moving on to the next section let’s work a couple of examples to remind us once again of some of the interpretations of the derivative. The reason for factoring the derivative will be apparent shortly. You appear to be on a device with a "narrow" screen width (, Derivatives of Exponential and Logarithm Functions, L'Hospital's Rule and Indeterminate Forms, Substitution Rule for Indefinite Integrals, Volumes of Solids of Revolution / Method of Rings, Volumes of Solids of Revolution/Method of Cylinders, Parametric Equations and Polar Coordinates, Gradient Vector, Tangent Planes and Normal Lines, Triple Integrals in Cylindrical Coordinates, Triple Integrals in Spherical Coordinates, Linear Homogeneous Differential Equations, Periodic Functions & Orthogonal Functions, Heat Equation with Non-Zero Temperature Boundaries, Absolute Value Equations and Inequalities, $$\displaystyle {\left( {f\left( x \right) \pm g\left( x \right)} \right)^\prime } = f'\left( x \right) \pm g'\left( x \right)\hspace{0.25in} \mbox{OR} \hspace{0.25in}\frac{d}{{dx}}\left( {f\left( x \right) \pm g\left( x \right)} \right) = \frac{{df}}{{dx}} \pm \frac{{dg}}{{dx}}$$, $$\displaystyle {\left( {cf\left( x \right)} \right)^\prime } = cf'\left( x \right)\hspace{0.25in} \mbox{OR} \hspace{0.25in}\frac{d}{{dx}}\left( {cf\left( x \right)} \right) = c\frac{{df}}{{dx}}$$, $$c$$ is any number, If $$f\left( x \right) = c$$ then $$\displaystyle f'\left( x \right) = 0\hspace{0.25in} \mbox{OR} \hspace{0.25in}\frac{d}{{dx}}\left( c \right) = 0$$.

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