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Chapter 7: Problem 31

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

Short Answer

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

Step by step solution

01

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: \( f(t)=(t+1)^{0.5}\).
02

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: \( f'(t)=0.5*(t+1)^{(0.5-1)}*1 \).
03

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 \( f'(t)=0.5/((t+1)^{0.5}) = 0.5/\sqrt{t+1} \).

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