Question

answer plzz you have a good day and I have to go to the store​

Answer 1

Question:-

1. Find the adjoint matrix of $\sf \left[\begin{array}{c c} sin\theta \: \: 1 \\ 0 \: \: cos\theta\end{array}\right]$

2. Find the derivative of the inverse function of the following:-

$\sf y = x.7^x$

Answer:-

1.

The adjoint of a matrix is transpose of the co-factor matrix.

$\sf M = \left[\begin{array}{c c} a \: \: b \\ c \: \: d\end{array}\right]$

$\sf adj.[M] = \left[\begin{array}{c c} d \: \: -b \\ -c \: \: \: a\end{array}\right]$

Similarly,

$\sf [M] = \left[\begin{array}{c c} sin\theta \: \: 1 \\ 0 \: \: cos\theta\end{array}\right]$

$\bf adj.[M] = \left[\begin{array}{c c} cos\theta \: \: -0 \\ -1 \: \: sin\theta\end{array}\right]$

$\implies$$\bf\green{adj.[M] = \left[\begin{array}{c c} cos\theta \: \: \infty \\ -1 \: \: sin\theta\end{array}\right]}\dashrightarrow\bf\red{[\because -0 = \infty]}$

2.

Derivative of $\sf y = x.7^x$

Differentiating with respect to x:-

→ $\sf \dfrac{dy}{dx} = \dfrac{d}{dx}(x.7^x)$

→ $\sf x \dfrac{d}{dx} [7^x] + 7^x \dfrac{d}{dx} (x)$

→ $\sf x.7^x log7 + 7^x \times 1$

→ $\sf 7x(x log7 +1)$

Derivative of inverse function:-

$\bf y = f(x) \implies \dfrac{dy}{dx} = \dfrac{1}{\bigg[\dfrac{dy}{dx}\bigg]}$

$\implies$$\bf\green{\dfrac{1}{7^x(x log7 + 1)}}$