Find the derivative of the given function.
$g\left(t\right)=\log_5\left(7^{t^2+4}\right)$

• A. $\frac{2t\ln7}{\ln5}$
• B. $2t\ln\frac75$
• C. $\frac{2t\ln7}{t\ln5}$
• D. $\frac{7^{t^{2+4}}}{t\ln5}$

Answer: (A)

Verified step by step guidance

1. Step 1: Recognize that the function g(t) = log_5(7^(t^2 + 4)) involves a logarithm with a base other than e or 10. To differentiate this, we will use the change of base formula: log_a(x) = ln(x) / ln(a).
2. Step 2: Rewrite g(t) using the change of base formula: g(t) = ln(7^(t^2 + 4)) / ln(5). This allows us to work with natural logarithms, which are easier to differentiate.
3. Step 3: Simplify ln(7^(t^2 + 4)) using the logarithmic property ln(a^b) = b * ln(a). This gives g(t) = [(t^2 + 4) * ln(7)] / ln(5).
4. Step 4: Differentiate g(t) with respect to t. Use the quotient rule for derivatives: if h(t) = f(t) / g(t), then h'(t) = [f'(t)g(t) - f(t)g'(t)] / [g(t)]^2. Here, f(t) = (t^2 + 4) * ln(7) and g(t) = ln(5).
5. Step 5: Since ln(5) is a constant, its derivative is 0. Focus on differentiating f(t): f'(t) = d/dt[(t^2 + 4) * ln(7)]. Use the product rule: if u(t) = t^2 + 4 and v(t) = ln(7), then f'(t) = u'(t)v(t) + u(t)v'(t). Substitute the results into the quotient rule to find g'(t).