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Problem 301

# Given $\mathrm{A}(\mathrm{t})=\begin{array}{rr}\mid \mathrm{t}^{2} & \cos \mathrm{t} \mid \\ \mid \mathrm{e}^{\mathrm{t}} & \sin \mathrm{t} \mid\end{array}$ Find $$(\mathrm{d} \mathrm{A} / \mathrm{dt})$$

Expert verified
The short answer to the question is: $$\frac{dA}{dt} = \begin{pmatrix} 2t & -\sin(t) \\ e^t & \cos(t) \end{pmatrix}$$
See the step by step solution

## Step 1: Recall the formula for finding the derivative of a matrix

To find the derivative of a matrix, we simply need to find the derivative of each element of the matrix with respect to the variable (in this case, t) and place the derivatives in the resulting matrix in their corresponding positions.

## Step 2: Find the derivatives of the matrix elements

We will find the derivatives of each element in the matrix A(t): 1. Derivative of $$t^2$$: using basic power rule: $$\frac{d}{dt}(t^2) = 2t$$ 2. Derivative of $$\cos(t)$$: using basic differentiation rules: $$\frac{d}{dt}(\cos(t)) = -\sin(t)$$ 3. Derivative of $$e^t$$: using basic exponential function differentiation: $$\frac{d}{dt}(e^t) = e^t$$ 4. Derivative of $$\sin(t)$$: using basic differentiation rules: $$\frac{d}{dt}(\sin(t)) = \cos(t)$$

## Step 3: Form the resulting matrix

Now that we have found the derivatives of each element of A(t), we can form the resulting matrix (dA/dt): $\frac{dA}{dt} = \begin{pmatrix} 2t & -\sin(t) \\ e^t & \cos(t) \end{pmatrix}$ That's it! The derivative of the given matrix A(t) is: $\frac{dA}{dt} = \begin{pmatrix} 2t & -\sin(t) \\ e^t & \cos(t) \end{pmatrix}$

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