# differential calculus examples

Example 2: f(x) = x 3. Make sure that you can deal with fractional exponents. Get help with your Differential calculus homework. It defines the ratio of the change in the value of a function to the change in the independent variable. The first two restrict the formula to $$n$$ being an integer because at this point that is all that we can do at this point. For example, velocity is the rate of change of distance with respect to time in a particular direction. We will introduce most of these formulas over the course of the next several sections. The only way that we’ll know for sure which direction the object is moving is to have the velocity in hand. Integral Calculus joins (integrates) the small pieces together to find how much there is. $$f\left( x \right) = 15{x^{100}} - 3{x^{12}} + 5x - 46$$, $$g\left( t \right) = 2{t^6} + 7{t^{ - 6}}$$, $$y = 8{z^3} - \frac{1}{{3{z^5}}} + z - 23$$, $$\displaystyle T\left( x \right) = \sqrt x + 9\sqrt{{{x^7}}} - \frac{2}{{\sqrt{{{x^2}}}}}$$, $$h\left( x \right) = {x^\pi } - {x^{\sqrt 2 }}$$, $$y = \sqrt{{{x^2}}}\left( {2x - {x^2}} \right)$$, $$\displaystyle h\left( t \right) = \frac{{2{t^5} + {t^2} - 5}}{{{t^2}}}$$. Don’t forget to do any basic arithmetic that needs to be done such as any multiplication and/or division in the coefficients. There is a general rule about derivatives in this class that you will need to get into the habit of using. Again, remember that the Power Rule requires us to have a variable to a number and that it must be in the numerator of the term. Here is the function written in “proper” form. Let’s graph these points on a number line. Now that we’ve gotten the function rewritten into a proper form that allows us to use the Power Rule we can differentiate the function. Khan Academy is a 501(c)(3) nonprofit organization. We can simplify this rational expression however as follows. Now in this function the second term is not correctly set up for us to use the power rule. When you see radicals you should always first convert the radical to a fractional exponent and then simplify exponents as much as possible. f '(x) = c g … In the first section of this chapter we saw the definition of the derivative and we computed a couple of derivatives using the definition. In other words, we can “factor” a multiplicative constant out of a derivative if we need to. We can see from the factored form of the derivative that the derivative will be zero at $$t = 2$$ and $$t = 5$$. In this function we can’t just differentiate the first term, differentiate the second term and then multiply the two back together. It’s a very common mistake to bring the 3 up into the numerator as well at this stage. To check the instantaneous rate of change such as velocity. It is still possible to do this derivative however. The derivative of a constant is zero. Here is the function written in “proper” form. It measures the steepness of the graph of a function. <> Now, we need to determine where the derivative is positive and where the derivative is negative. Differential calculus deals with the rate of change of one quantity with respect to another. For problems 1 – 12 find the derivative of the given function. 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. There are actually three different proofs in this section. In the first section of this chapter we saw the definition of the derivative and we computed a couple of derivatives using the definition. The analytical tutorials may be used to further develop your skills in solving problems in calculus. so be careful with this! Determine when the object is moving to the right and when the object is moving to the left. That doesn’t mean that we can’t differentiate any product or quotient at this point. The reason for factoring the derivative will be apparent shortly. 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. All of the terms in this function have roots in them. We included negative $$t$$’s here because we could even though they may not make much sense for this problem. For example, velocity and slopes of tangent lines. You will be asked numerous times over the course of the next two chapters to determine where functions are positive and/or negative. In all of the previous examples the exponents have been nice integers or fractions. %PDF-1.4 Recall that if the velocity is positive the object is moving off to the right and if the velocity is negative then the object is moving to the left. Independent variables are the inputs to the functions that define the quantity which is being manipulated in an experiment. Luckily for us we won’t have to use the definition terribly often. y = 2t4 −10t2+13t y = 2 t 4 − 10 t 2 + 13 t Solution. It’s often easier to do the evaluation with positive exponents. If we can first do some simplification the functions will sometimes simplify into a form that can be differentiated using the properties and formulas in this section. The derivative of a product or quotient of two functions is not the product or quotient of the derivatives of the individual pieces. The study of differential calculus is concerned with how one quantity changes in relation to another quantity. The reason for factoring the derivative will be apparent shortly. See the Proof of Various Derivative Formulas section of the Extras chapter to see the proof of this formula. In the last two terms we combined the exponents. Solution: We know, $$\frac{\mathrm{d} (x^n)}{\mathrm{d} x}$$ = n x n-1 That is usually what we’ll see in this class. However, this problem is not terribly difficult it just looks that way initially. As with the first part we can’t just differentiate the numerator and the denominator and the put it back together as a fraction. Required fields are marked *. In differential calculus basics, you may have learned about differential equations, derivatives, and applications of derivatives. Differential calculus deals with the study of the rates at which quantities change. Note that we have not included formulas for the derivative of products or quotients of two functions here. So, as we saw in this example there are a few products and quotients that we can differentiate. See the Proof of Various Derivative Formulas section of the Extras chapter to see the proof of this formula. "X#�G�ҲR(� F#�{� ����wY�ifT���o���T/�.~5�䌖���������|]��:� �������B3��0�d��Aڣh�4�t���.��Z �� We will give the properties and formulas in this section in both “prime” notation and “fraction” notation. The object is moving to the right and left in the following intervals. Now that we’ve gotten the function rewritten into a proper form that allows us to use the Power Rule we can differentiate the function. You should always do this with this kind of term. The method that we tend to prefer is the following. So, at $$x = - 2$$ the derivative is negative and so the function is decreasing at $$x = - 2$$. The central concept of differential calculus is the derivative. 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. A brief introduction to differential calculus. Following this rule will save you a lot of grief in the future. Therefore, all that we need to do is to check the derivative at a test point in each region and the derivative in that region will have the same sign as the test point. We can now differentiate the function. y = √x +8 3√x −2 4√x y = x + 8 x 3 − 2 x 4 Solution. Definition of Derivative: The following formulas give the Definition of Derivative. 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). The point of this problem is to make sure that you deal with negative exponents correctly. In each of these regions we know that the derivative will be the same sign. So, upon evaluating the derivative we get. So, as we saw in this example there are a few products and quotients that we can differentiate. There is a general rule about derivatives in this class that you will need to get into the habit of using. In order to use the power rule we need to first convert all the roots to fractional exponents. So, prior to differentiating we first need to rewrite the second term into a form that we can deal with. The derivative of f(x) = c where c is a constant is given by f '(x) = 0 Example f(x) = - 10 , then f '(x) = 0 2 - Derivative of a power function (power rule). That is usually what we’ll see in this class. Interpretation of the Derivative as the Slope of a Tangent.