15–48. Derivatives Find the derivative of the following functions.
y = 5^3t

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Step 1: Identify the function type. The given function is y = 5^(3t), which is an exponential function where the base is a constant and the exponent is a linear function of t.

Step 2: Recall the derivative rule for exponential functions. If you have a function of the form y = a^(u(t)), where a is a constant and u(t) is a function of t, the derivative is given by y' = a^(u(t)) * ln(a) * u'(t).

Step 3: Apply the derivative rule. In this case, a = 5 and u(t) = 3t. Therefore, the derivative y' = 5^(3t) * ln(5) * (d/dt)(3t).

Step 4: Compute the derivative of the exponent. The derivative of u(t) = 3t with respect to t is simply 3, since the derivative of t is 1 and 3 is a constant multiplier.

Step 5: Combine the results. Substitute the derivative of the exponent back into the formula: y' = 5^(3t) * ln(5) * 3.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivatives

A derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus that allows us to determine how a function behaves as its input changes. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a given point.

Exponential Functions

Exponential functions are mathematical functions of the form y = a^x, where 'a' is a constant and 'x' is the variable. In the context of derivatives, the derivative of an exponential function can be calculated using specific rules, such as the fact that the derivative of a^x is a^x * ln(a). Understanding how to differentiate exponential functions is crucial for solving problems involving them.

Power Rule

The power rule is a basic rule for finding the derivative of functions of the form y = x^n, where 'n' is a real number. According to the power rule, the derivative is given by dy/dx = n*x^(n-1). This rule simplifies the process of differentiation, especially for polynomial and power functions, and is essential for solving a wide range of calculus problems.

15–48. Derivatives Find the derivative of the following functions.y = 5^3t