Question

Differentiate r(t) = [(t+1)^(-1), atan(t), ln(t+1)] with respect to t.

Answer 1

To differentiate r(t) = [(t+1)^(-1), atan(t), ln(t+1)] with respect to t, we find the derivatives of each component separately using the appropriate rules. The derivative of r(t) with respect to t is [(t+1)^(-2), 1/(1+t^2), 1/(t+1)].

Step-by-step explanation:

To differentiate r(t) = [(t+1)^(-1), atan(t), ln(t+1)] with respect to t, we will differentiate each component of r(t) separately. Let's denote the three components of r(t) as x(t), y(t), and z(t).

1. To find the derivative of x(t), we use the power rule: (t+1)^(-1) differentiates to -1*(t+1)^(-2) which simplifies to -1/(t+1)^2.
2. To find the derivative of y(t), we use the derivative of atan(t) which is 1/(1+t^2).
3. To find the derivative of z(t), we use the derivative of ln(t+1) which is 1/(t+1).

Therefore, the derivative of r(t) with respect to t is [(t+1)^(-2), 1/(1+t^2), 1/(t+1)].