91Ó°ÊÓ

The chain rule is a fundamental technique in calculus for differentiating compositions of functions. It essentially states that to find the derivative of a composition, you must first differentiate the outer function and then multiply it by the derivative of the inner function.

In the example provided, we need to differentiate the function \(f(t) = 5^{t+1}\). This functions involves an exponent of \((t + 1)\), making it a candidate for the chain rule. The outer function here is the exponential base \(5\), and the inner function is \(t + 1\).

Using the chain rule:

• Differentiate the outer function: the derivative of \(5^u\) with respect to \(u\) is \(5^u \ln(5)\).
• Multiply by the derivative of the inner function \(t+1\), which is simply 1.

This gives us the first derivative: \(f'(t) = 5^{t+1} \ln(5)\), effectively applying the chain rule.

A second derivative provides information on the curvature or the concavity of the original function. While the first derivative tells us about the slope and direction in which a function is heading, the second derivative tells us how that slope itself is changing over time.

In our exercise, we found the first derivative, \(f'(t) = 5^{t+1} \ln(5)\), which is again an exponential function times a constant. To find the second derivative \(f''(t)\), we differentiate \(f'(t)\) again. Applying the chain rule here, differentiate \(5^{t+1} \ln(5)\):

• Use the chain rule on \(5^{t+1}\), same as before, giving \(5^{t+1} \ln(5)\) as the derivative for \(5^{t+1}\).
• Multiply by the constant \(\ln(5)\) again, due to its presence in the first derivative.

This results in the second derivative: \(f''(t) = 5^{t+1} \ln^2(5)\). The pattern emerges that each differentiation involves multiplying again by \(\ln(5)\).

Exponential functions are a powerful concept in calculus, characterized by their constant proportional growth rates. They take the form \(a^x\), where \(a\) is a positive real number and \(x\) is any real number. Exponential functions are unique due to their rapid increase or decrease, depending on the base.

In our exercise, the function \(f(t) = 5^{t+1}\) is an exponential function with base 5 and can be understood as \(g(u) = 5^u\) where \(u = t + 1\). Since the base is greater than 1, this indicates exponential growth.

Derivatives of exponential functions continue to hold onto their exponential nature:

• Each derivative multiplies by the natural logarithm of the base, \(\ln(5)\) in this case.
• The nature of the exponential part \(5^{t+1}\) remains unchanged across derivatives.

This is why the chain rule becomes so intuitive for these functions—each differentiation respects the original exponential structure, just tweaking the coefficients depending on the order of the derivative taken.