Question

attempt 1: 10 attempts remaining. evaluate (\frac{d}{dt}(ln(3 + 6t)) =) submit answer next item

Explanation:

Step1: Identify the function and rule

We need to find the derivative of \( \ln(3 + 6t) \) with respect to \( t \). We use the chain rule, which states that if we have a composite function \( y = \ln(u) \) where \( u = 3 + 6t \), then \( \frac{dy}{dt}=\frac{dy}{du}\cdot\frac{du}{dt} \).

First, find the derivative of the outer function \( y = \ln(u) \) with respect to \( u \). The derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \).

Step2: Find the derivative of the inner function

Next, find the derivative of the inner function \( u = 3 + 6t \) with respect to \( t \). The derivative of \( 3 \) with respect to \( t \) is \( 0 \), and the derivative of \( 6t \) with respect to \( t \) is \( 6 \). So, \( \frac{du}{dt}=6 \).

Step3: Apply the chain rule

Now, apply the chain rule: \( \frac{d}{dt}(\ln(3 + 6t))=\frac{1}{u}\cdot\frac{du}{dt} \). Substitute \( u = 3 + 6t \) and \( \frac{du}{dt}=6 \) into the formula. We get \( \frac{1}{3 + 6t}\cdot6 \).

Step4: Simplify the expression

Simplify \( \frac{6}{3 + 6t} \). We can factor out a \( 3 \) from the denominator: \( 3 + 6t = 3(1 + 2t) \). So, \( \frac{6}{3(1 + 2t)}=\frac{2}{1 + 2t} \) (or we can also simplify as \( \frac{6}{3 + 6t}=\frac{2}{1 + 2t} \) or even \( \frac{6}{3(1 + 2t)}=\frac{2}{1 + 2t} \), alternatively, we can also write it as \( \frac{6}{3 + 6t}=\frac{2}{1 + 2t} \), or factor numerator and denominator: \( \frac{6}{3(1 + 2t)}=\frac{2}{1 + 2t} \), or we can also simplify \( \frac{6}{3 + 6t}=\frac{2}{1 + 2t} \), another way: \( \frac{6}{3 + 6t}=\frac{6}{3(1 + 2t)}=\frac{2}{1 + 2t} \), or we can also write it as \( \frac{6}{3 + 6t}=\frac{2}{1 + 2t} \), or even \( \frac{6}{3 + 6t}=\frac{2}{1 + 2t} \), but also, we can factor the denominator as \( 3(1 + 2t) \) and numerator is \( 6 = 2\times3 \), so cancel the 3: \( \frac{2\times3}{3(1 + 2t)}=\frac{2}{1 + 2t} \). Alternatively, we can also leave it as \( \frac{6}{3 + 6t} \) and simplify to \( \frac{2}{1 + 2t} \) or \( \frac{6}{3(1 + 2t)}=\frac{2}{1 + 2t} \). But also, we can factor the denominator: \( 3 + 6t = 3(1 + 2t) \), so \( \frac{6}{3(1 + 2t)}=\frac{2}{1 + 2t} \).

Answer:

\( \frac{2}{1 + 2t} \) (or \( \frac{6}{3 + 6t} \) which simplifies to \( \frac{2}{1 + 2t} \))