Question

What is the derivative of \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle ?\)

Short answer

Question: Determine the derivative of the vector-valued function \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) with respect to \(t\). Answer: The derivative of the vector-valued function \(\mathbf{r}(t)\) with respect to \(t\) is \(\frac{d\mathbf{r}(t)}{dt} = \langle f'(t), g'(t), h'(t) \rangle\).

Step-by-step solution

Identify the components of the vector-valued function

The given vector-valued function is \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\). The components of this function are \(f(t)\), \(g(t)\), and \(h(t)\).

Differentiate each component with respect to t

To find the derivative of the vector-valued function, we need to differentiate each component with respect to \(t\). Let's denote the derivatives as \(f'(t)\), \(g'(t)\), and \(h'(t)\). These represent the derivatives of \(f(t)\), \(g(t)\), and \(h(t)\) with respect to \(t\), respectively.

Write the derivative in vector form

Now that we have found the derivatives of each component with respect to \(t\), we can write the derivative of the vector-valued function in vector form. The derivative \(\frac{d\mathbf{r}(t)}{dt}\) is given by: $$ \frac{d\mathbf{r}(t)}{dt} = \langle f'(t), g'(t), h'(t) \rangle. $$