Prove derivative formula

5 Prove by mathematical induction that, for every positive integer \(n\), $$\frac { d ^ { 2 n - 1 } } { d x ^ { 2 n - 1 } } ( x \sin x ) = ( - 1 ) ^ { n - 1 } ( x \cos x + ( 2 n - 1 ) \sin x )$$

2 It is given that \(\mathrm { y } = \mathrm { xe } ^ { \mathrm { ax } }\), where \(a\) is a constant.
Prove by mathematical induction that, for all positive integers \(n\), $$\frac { d ^ { n } y } { d x ^ { n } } = \left( a ^ { n } x + n a ^ { n - 1 } \right) e ^ { a x }$$

4 The function f is such that \(\mathrm { f } ^ { \prime \prime } ( x ) = \mathrm { f } ( x )\).
Prove by mathematical induction that, for every positive integer \(n\), $$\frac { d ^ { 2 n - 1 } } { d x ^ { 2 n - 1 } } ( x f ( x ) ) = x f ^ { \prime } ( x ) + ( 2 n - 1 ) f ( x )$$

2 Prove by mathematical induction that, for all positive integers \(n\), $$\frac { d ^ { n } } { d x ^ { n } } \left( x ^ { 2 } e ^ { x } \right) = \left( x ^ { 2 } + 2 n x + n ( n - 1 ) \right) e ^ { x }$$

3 Prove by mathematical induction that, for every positive integer \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \mathrm { e } ^ { x } \sin x \right) = ( \sqrt { } 2 ) ^ { n } \mathrm { e } ^ { x } \sin \left( x + \frac { 1 } { 4 } n \pi \right)$$

2 Prove by mathematical induction that, for all positive integers \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \frac { 1 } { 2 x + 3 } \right) = ( - 1 ) ^ { n } \frac { n ! 2 ^ { n } } { ( 2 x + 3 ) ^ { n + 1 } }$$

3

1. Show that \(\frac { \mathrm { d } ^ { n + 1 } } { \mathrm {~d} x ^ { n + 1 } } \left( x ^ { n + 1 } \ln x \right) = \frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( x ^ { n } + ( n + 1 ) x ^ { n } \ln x \right)\).
2. Prove by mathematical induction that, for all positive integers \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( x ^ { n } \ln x \right) = n ! \left( \ln x + 1 + \frac { 1 } { 2 } + \ldots + \frac { 1 } { n } \right)$$

2 It is given that \(y = \ln ( a x + 1 )\), where \(a\) is a positive constant. Prove by mathematical induction that, for every positive integer \(n\), $$\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } } = ( - 1 ) ^ { n - 1 } \frac { ( n - 1 ) ! a ^ { n } } { ( a x + 1 ) ^ { n } }$$

3 Given that \(a\) is a constant, prove by mathematical induction that, for every positive integer \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( x \mathrm { e } ^ { a x } \right) = n a ^ { n - 1 } \mathrm { e } ^ { a x } + a ^ { n } x \mathrm { e } ^ { a x }$$

3 Prove by induction that, for all \(n \geqslant 1\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \mathrm { e } ^ { x ^ { 2 } } \right) = \mathrm { P } _ { n } ( x ) \mathrm { e } ^ { x ^ { 2 } } ,$$ where \(\mathrm { P } _ { n } ( x )\) is a polynomial in \(x\) of degree \(n\) with the coefficient of \(x ^ { n }\) equal to \(2 ^ { n }\).

3 Prove by mathematical induction that, for all positive integers \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \mathrm { e } ^ { x } \sin x \right) = 2 ^ { \frac { 1 } { 2 } n } \mathrm { e } ^ { x } \sin \left( x + \frac { 1 } { 4 } n \pi \right)$$

5 It is given that \(y = ( 1 + x ) ^ { 2 } \ln ( 1 + x )\). Find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\). Prove by mathematical induction that, for every integer \(n \geqslant 3\), $$\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } } = ( - 1 ) ^ { n - 1 } \frac { 2 ( n - 3 ) ! } { ( 1 + x ) ^ { n - 2 } }$$

6 Let \(\mathrm { y } = \mathrm { x } \cosh \mathrm { x }\).
Prove by induction that, for all integers \(n \geqslant 1 , \frac { d ^ { 2 n - 1 } y } { d x ^ { 2 n - 1 } } = x \sinh x + ( 2 n - 1 ) \cosh x\).

Prove by mathematical induction that, for all positive integers \(n\), $$\frac{\mathrm{d}^n}{\mathrm{d}x^n}\left(\tan^{-1}x\right) = P_n(x)\left(1+x^2\right)^{-n},$$ where \(P_n(x)\) is a polynomial of degree \(n-1\). [6]

Given that \(y = x \sin x\), find \(\frac{d^2y}{dx^2}\) and \(\frac{d^4y}{dx^4}\), simplifying your results as far as possible, and show that $$\frac{d^6y}{dx^6} = -x \sin x + 6 \cos x.$$ [3] Use induction to establish an expression for \(\frac{d^{2n}y}{dx^{2n}}\), where \(n\) is a positive integer. [5]

Given that \(a\) is a constant, prove by mathematical induction that, for every positive integer \(n\), $$\frac{\mathrm{d}^n}{\mathrm{d}x^n}(xe^{ax}) = na^{n-1}e^{ax} + a^n xe^{ax}.$$ [6]

It is given that \(y = e^x u\), where \(u\) is a function of \(x\). The \(r\)th derivatives \(\frac{\mathrm{d}^r y}{\mathrm{d}x^r}\) and \(\frac{\mathrm{d}^r u}{\mathrm{d}x^r}\) are denoted by \(y^{(r)}\) and \(u^{(r)}\) respectively. Prove by mathematical induction that, for all positive integers \(n\), $$y^{(n)} = e^x\left[\binom{n}{0}u + \binom{n}{1}u^{(1)} + \binom{n}{2}u^{(2)} + \ldots + \binom{n}{r}u^{(r)} + \ldots + \binom{n}{n}u^{(n)}\right].$$ [8] [You may use without proof the result \(\binom{k}{r} + \binom{k}{r-1} = \binom{k+1}{r}\).]

It is given that \(y = \ln(ax + 1)\), where \(a\) is a positive constant. Prove by mathematical induction that, for every positive integer \(n\), $$\frac{d^n y}{dx^n} = (-1)^{n-1} \frac{(n-1)!a^n}{(ax+1)^n}.$$ [6]

It is given that \(y = \ln(ax + 1)\), where \(a\) is a positive constant. Prove by mathematical induction that, for every positive integer \(n\), $$\frac{d^n y}{dx^n} = (-1)^{n-1} \frac{(n-1)! a^n}{(ax+1)^n}.$$ [6]