We study different power rules in calculus which are used in differentiation, integration, for simplifying exponents and logarithmic functions. Now, we will find the derivative of g(x) using the power rule. Consider the polynomial g(x) = x -4 + x 3/4 -7x 1/9 + 3. Let us now consider an algebraic expression with rational exponents and apply the formula to find its derivative. Hence, we have determined the derivative of a polynomial including terms of the form x n using the power rule. To find the derivative of f(x) = 3x 4 - 2x -2 + x - 8, we will first use the fact that the derivative of sum of functions is equal to the sum of the derivatives of the functions, that is, (u + v)' = u' + v'. We will find the derivative of f(x) = 3x 4 - 2x -2 + x - 8 using the power rule formula and in this expression the power of x is both positive and negative. Now, in this section, we understand how to apply the power rule for the differentiation of algebraic expressions (or polynomials) including terms of the form x n. Hence, we can conclude that the formula d(x n)/dx = nx n-1 is true for all real numbers n. ![]() Similarly, we can use the implicit differentiation method to prove the formula for rational exponents as well. Hence, for all negative integers, the power rule formula d(x n)/dx = nx n-1 holds true. Assume n = -m, where m is a natural number. Next, we will generalize the power rule formula for negative integers. Power Rule Formula Proof for Negative Integers Hence, we have proved the power rule formula for differentiation for positive integers n. According to the first principle, the derivative of f(x) = x n is given by, ![]() We will use the first principle of differentiation to prove the formula and hence, use the binomial formula to arrive at the result. The formula for binomial theorem is given by, (x + y) n = nC 0 x n + nC 1 x n-1 y + nC 2 x n-2 y 2 + nC 3 x n-3 y 3 + nC 4 x n-4 y 4 +. In this section, we will prove the general power rule formula for differentiation using the binomial theorem formula. Proof of Power Rule Using Binomial Theorem Therefore, using the principle of mathematical induction, we have proved that P(n): d(x n)/dx = nx n-1 is true for all natural numbers, and hence, we have proved the power rule formula for differentiation. Step 3: Now, we will prove P(n) is true for n = k + 1, that is, we need to prove d(x k+1)/dx = (k+1)x k Step 2: Assume P(k) is true, that is, d(x k)/dx = kx k-1 - (2) Step 1: Assume n = 1, then we have LHS = dx/dx = 1 (Because derivative of x is equal to 1). Then, assume P(n) to be true for n = k, we will prove it for n = k+1. Here our statement is P(n): d(x n)/dx = nx n-1. Using the principle of mathematical induction, we will prove the formula d(x n)/dx = nx n-1 for positive integral values of n. Power Rule Proof Using Mathematical Induction We will prove the general formula for the power rule using the principle of mathematical induction and the binomial theorem. ![]() Now we know the formula for the power rule derivative, let us now prove the formula using different methods.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |