Pregunta
Find the derivative of the function. \[ f(t)=66^{t^{3}} \]
Ask by Daniel Ramos. in the United States
Mar 17,2025
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The derivative of \( f(t) = 66^{t^3} \) is \( f'(t) = 3t^2 \ln(66) \cdot 66^{t^3} \).
Solución
Let \( f(t)=66^{t^3} \). We can differentiate this function by using the general formula for the derivative of an exponential function with a non-constant exponent:
\[
\frac{d}{dt}\left(a^{g(t)}\right)=a^{g(t)}\ln(a)\cdot g'(t)
\]
In this case, we have:
- \( a=66 \)
- \( g(t)=t^3 \)
First, compute \( g'(t) \):
\[
g'(t)=\frac{d}{dt}\left(t^3\right)=3t^2
\]
Now, applying the formula:
\[
f'(t)=66^{t^3}\ln(66)\cdot 3t^2
\]
Thus, the derivative of the function is:
\[
f'(t)=3t^2\ln(66) \cdot 66^{t^3}
\]
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Simplifique esta solución Extra Insights
To find the derivative of the function \( f(t) = 66^{t^3} \), we can use the chain rule and the property of exponential functions. Recall that the derivative of \( a^{u} \) with respect to \( t \) is given by \( a^{u} \ln(a) \frac{du}{dt} \). Here, we have \( a = 66 \) and \( u = t^3 \). First, we compute the derivative of \( u \): \[ \frac{du}{dt} = 3t^2. \] Now, applying the derivative rule: \[ f'(t) = 66^{t^3} \ln(66) \cdot 3t^2. \] Thus, the derivative is: \[ f'(t) = 3t^2 \cdot 66^{t^3} \ln(66). \]
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