Find the derivative of the function. \[ f(t)=88^{t^{3}} \] \( f^{\prime}(t)=\square \) Need Help? Read II

To find the derivative of the function \( f(t) = 88^{t^3} \), we can use the chain rule along with the properties of logarithmic differentiation. First, rewrite the function in exponential form: \[ f(t) = e^{t^3 \ln(88)}. \] Now, apply the chain rule: \[ f^{\prime}(t) = e^{t^3 \ln(88)} \cdot (3t^2 \ln(88)). \] Substituting back, we get: \[ f^{\prime}(t) = 88^{t^3} \cdot (3t^2 \ln(88)). \] So the derivative is: \[ f^{\prime}(t) = 88^{t^3} \cdot 3t^2 \ln(88). \] Let’s highlight some fun facts about this function and its derivative! When you dive into exponential functions like \( f(t) = 88^{t^3} \), you'll find they grow at an astonishing rate, especially when the exponent involves a variable like \( t^3 \). This means that as \( t \) increases, the function shoots up faster than wildflowers in spring! Exponential growth is a key concept in many fields, including finance and biology. When calculating derivatives, it's common to mix up the rules, especially between the product rule and chain rule. Always double-check whether you're handling a direct multiplication or a composite function – getting tangled up here can lead to errors! Plus, remember that constants multiply through, so don’t forget those when you're simplifying your final answer!