Understanding the Process of Reversing Differentiation
Reversing the differentiation process involves finding the function knowing its derivative. By increasing the exponent by 1, dividing by the new exponent, and adding a constant term, the antiderivative of a function can be determined. The general form of an antiderivative involves adding a constant, making it flexible for various functions. Examples and explanations of antiderivatives are provided to enhance understanding and application in calculus.
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7 October 2024 Reversing the process of differentiation LO: To find the function knowing the derivative of the function. www.mathssupport.org
Reversing the process of differentiation When differentiating powers of x, y = xn, we multiply by the original exponent and then drop the exponent by one. We can write this process as follows: nxn reduce the exponent by 1 multiply by the the original exponent xn nxn-1 Reversing the process of differentiation given above would give nxn xn increase the exponent by 1 divide by the exponent nxn-1 dy dx Suppose we are given the derivative = xnand asked to find y in terms of x. increase the exponent by 1 xn+1 ??+1 ? + 1 divide by the exponent xn www.mathssupport.org
Reversing the process of differentiation The process of finding a function f(x) whose derivative is f (x) is called anti-differentiation, which relates to integration. For example, What is the derivative of x2 with respect to x? If you anti-differentiate 2xwith respect to x 2x x2 What is the derivative of x2 + 3with respect to x? What is the derivative of x2 5with respect to x? You can easily see that 2xis the derivative for any function of the formx2 + c, where c is any real number The antiderivative of all of them is the same 2x 2x This set, or family, of all anti-derivatives of a function is called the indefinite integral We therefore have to write y = x2 + c. www.mathssupport.org
Reversing the process of differentiation General form of an Antiderivative Let Fbe an antiderivative of f over an interval I, then i. For each constant C, the function F(x) + Cis also an antiderivative of f over I; ii. If G is an antiderivative of foverI, there is a constant C to which G(x) = F(x) + Cover I. In other words, the most general form of the antiderivative of fover I is F(x) + C www.mathssupport.org
Reversing the process of differentiation dy dx For example: Adding 1 to the power and dividing by the new power gives: 2 If =6 find in terms of . y x x 6 3 =3 = 2x3 y x This is not the complete solution, however, because if we differentiated y = 2x3 + 1, ory = 2x3 3, or y = 2x3 + any constant dy x dx we would also get 2 = 6 We therefore have to write y = 2x3 + c. In general: The antiderivative of xnare given by 1 ? + 1??+1+ ? www.mathssupport.org
Reversing the process of differentiation Find the antiderivative of the function x10: Adding 1 to the power and dividing by the new power gives: 1 10 + 1?10+1+ ? 1 11?11+ ? = Find the antiderivative of the function 1 Adding 1 to the power and dividing by the new power gives: 1 5 + 1? 5+1+ ? 1 4? 4+ ? = x-5 ?5 1 = 4?4+ ? = www.mathssupport.org
Reversing the process of differentiation Find the antiderivative of the function Rewriting using rational exponents 4?3 3 4 = ? Adding 1 to the power and dividing by the new power gives: 1 3 4+ 1 =1 7 4 =4 7? 3 4+1+ ? ? 7 4+ ? ? 7 4+ ? www.mathssupport.org
Reversing the process of differentiation Antidifferentiation is also known as indefinite integration And is denoted with an integral symbol, ?? IfF (x) = f(x), we write ? ? ?? = ? ? + ? Constant of integration integrand Variable of integration Is read as: The antiderivative of fwith respect to x The integral of fwith respect to x ? ? ?? www.mathssupport.org
Rules to find the indefinite integrals Power rule 1 ? + 1??+1+ ?,? 1 ???? = Constant rule ??? = ?? + ?, Constant multiple rule ??(?)?? = ? ?(?)?? Sum or difference rule = ? ? ?? ? ? ?? ?(?) ?(?)?? www.mathssupport.org
Indefinite integrals Find the indefinite integral Applying the power rule ?6?? 1 ? + 1??+1+ ?,? 1 ???? = 1 6 + 1?6+1+ ? =1 7?7+ ? Find the indefinite integral Applying the constant rule 4?? ??? = ?? + ?, Thedttells you that the variable of integration is t = 4? + ?, www.mathssupport.org
Indefinite integrals Find the indefinite integral Applying the constant multiple rule 3?5?? ??(?)?? = ? ?(?)?? = 3 ?5?? 1 5 + 1?5+1+ ? = 3 =1 2?6+ ? www.mathssupport.org
Indefinite integrals Find the indefinite integral Applying the sum or difference rule (3?4+6?2+ 2)?? ?(?) ?(?)?? = ? ? ?? ? ? ?? = 3?4?? + 6?2?? + 2?? Applying the constant multiple rule = 3 ?4?? +6 ?2?? + 2?? 1 1 Applying the power rule and constant rule 4 + 1?4+1 2 + 1?2+1 = 3 +6 +2? + ? =3 5?5+ 2?3+ 2? + ? www.mathssupport.org
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