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

Guida verificata passo dopo passo

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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