promath The nth derivative of xn is equal to n!. The nth derivative of x^n is denoted by $\frac{d^n}{dx^n}\left( x^n\right)$, and its formula is given as follows: $\boxed{\dfrac{d^n}{dx^n}\left( x^n\right)=n!}$ nth Derivative of xn Question: Find nth Derivative of xn. Answer: The nth derivative of x to the power n is obtained by repeatedly using the power … Read more
The nth derivative of 1/x is equal to (-1)nn!/xn+1. This is obtained by repeatedly using the power rule of differentiation. The nth derivative of 1/x is denoted by $\dfrac{d^n}{dx^n}\left( \dfrac{1}{x}\right)$, and its formula is given as follows: $\boxed{\dfrac{d^n}{dx^n}\left( \dfrac{1}{x}\right)=\dfrac{(-1)^n n!}{x^{n+1}}}$ nth Derivative of 1/x Question: How to find nth Derivative of 1/x? Answer: To find … Read more
The integral of xcosx is equal to xsinx +cosx+C where C is an arbitrary constant, and it is denoted by ∫xcosx dx. The function xcosx is a product of two functions x and cosx. So we can use integration by parts formula to find its integration. Notation of Integral of xcosx: ∫xcosx dx Integration formula … Read more
The limit of (x^n-a^n)/(x-a) as x approaches a is equal to nan-1. This limit is denoted by limx→a (xn-an)/(x-a), so the limit formula of (xn-an)/(x-a) when x tends to a is given as follows. limx→a (xn-an)/(x-a) = n⋅an-1 Lets prove this limit formula. Proof of limx→a (xn-an)/(x-a) To prove limx→a (xn-an)/(x-a) = n⋅an-1 we will consider three different cases … Read more
The limit of (x^n-1)/(x-1) as x approaches 1 is equal to n, that is, limx→1 (xn-1)/(x-1) = n. This follows from the formula limx→a (xn-an)/(x-a) = n⋅an-1 Put a=1, so we get that limx→1 $\dfrac{x^n-1}{x-1}$ = n. Let us now prove this limit formula. What is the Limit of (x^n-1)/(x-1) Answer: limx→1 (xn-1)/(x-1) is equal … Read more
The limit of 1/x^2 as x approaches infinity is equal to 0. As this limit is denoted by limx→∞ 1/x2, so the formula of the limit of 1/x2 is given as follows: limx→∞ 1/x2 = 0 What is the Limit of 1/x2 when x→∞ Answer: limx→∞ $\dfrac{1}{x^2}$ = 0. Explanation: As limx→∞ f(x)/g(x) is written as $\dfrac{\lim\limits_{x \to … Read more
The limit of x sin(1/x) as x approaches 0 is equal to 0. This limit is denoted by limx→0 xsin(1/x). So the formula for the limit of x sin(1/x) when x tends to zero is as follows. limx→0 x sin(1/x) = 0. Let us now find the limit of xsin(1/x) using the Squeeze/Sandwich theorem. Proof … Read more
The limit of x^2 sin(1/x) as x approaches 0 is equal to 0, and it is denoted by limx→0 x2 sin(1/x) = 0. So the limit formula of x2 sin(1/x) when x tends to zero is given by limx→0 x2 sin(1/x) = 0. We will now find the limit of x2 sin(1/x) using the Sandwich/Squeeze … Read more
The limit of x^1/x as x approaches infinity is equal to 1. This limit is denoted by limx→∞ x1/x, so the formula for the limit of x1/x when x tends to infinity is given by limx→∞ x1/x = 1. Let us now find the limit of x to the power 1/x when x tends to … Read more
The Laplace transform of e^4t is equal to 1/(s-4) and the Laplace of e^-4t is equal to 1/(s+4). This is because, we know that the Laplace of eat is 1/(s-a). The Laplace transform formulae for the functions e4t and e-4t are given in the table below. Function f(t) L{ f(t) } e4t L{e4t} = $\dfrac{1}{s-4}$ … Read more