Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Use the General Power Rule to find the derivative of the function. $$ f(t)=\sqrt{t+1} $$

Step-by-Step Solution

Answer

The derivative of the function \( f(t)=\sqrt{t+1} \) is \( f'(t)=0.5/\sqrt{t+1} \).

Step 1: Rewrite the Function

To start, rewrite the function \( f(t)=\sqrt{t+1} \) in a format where the power rule can be applied directly. It could be rewritten as: \( f(t)=(t+1)^{0.5}\).

Step 2: Apply the General Power Rule

After rewriting the function, apply the general power rule which states that if you have a function in the form of \( f(t)=g(t)^n \), then its derivative is: \( f'(t)=n*g(t)^{n-1}*g'(t)\). Here, \( g(t)=t+1 \), \( g'(t)=1 \), \( n=0.5 \). Replace \( g(t) \), \( n \) and \( g'(t) \) in the formula. The derivative of the function will be: \( f'(t)=0.5*(t+1)^{(0.5-1)}*1 \).

Step 3: Simplify the Solution

The next step is to simplify the expression above. The exponent \( 0.5-1=-0.5 \) and the function can be rewritten with a positive exponent as: \( f'(t)=0.5*(t+1)^{-0.5} \). This simplifies to \( f'(t)=0.5/((t+1)^{0.5}) = 0.5/\sqrt{t+1} \).