Video: Finding the Derivative of a Function Involving Trigonometric and Exponential Functions Using the Product Rule

Find the derivative of the function 𝑓(𝑑) = 𝑒^(π‘Žπ‘‘) sin 𝑏𝑑.

02:34

Video Transcript

Find the derivative of the function 𝑓 of 𝑑 is equal to 𝑒 to the power of π‘Žπ‘‘ times the sin of 𝑏𝑑.

The question gives us a function 𝑓 in terms of 𝑑. It wants us to find the derivative of this function. So, we want to differentiate this with respect to 𝑑. We can see that our function 𝑓 of 𝑑 is the product of two functions. It’s the product of an exponential function and a trigonometric function. And we know how to differentiate the product of two functions by using the product rule.

If we have that 𝑓 of 𝑑 is equal to the product of two functions 𝑒 of 𝑑 and 𝑣 of 𝑑, then the product rule tells us 𝑓 prime of 𝑑 is equal to 𝑣 of 𝑑 times 𝑒 prime of 𝑑 plus 𝑒 of 𝑑 times 𝑣 prime of 𝑑. So, to use the product rule, we’ll set our function 𝑒 of 𝑑 to be 𝑒 to the power of π‘Žπ‘‘, and we’ll set our function 𝑣 of 𝑑 to be the sin of 𝑏𝑑. To use the product rule, we’re going to need to find the expressions for 𝑒 prime of 𝑑 and 𝑣 prime of 𝑑.

Let’s start with 𝑒 prime of 𝑑. It is the derivative of 𝑒 to the power of π‘Žπ‘‘ with respect to 𝑑. And we know how to evaluate this derivative. We know for any constant 𝑛, the derivative of 𝑒 to the power of 𝑛𝑑 with respect to 𝑑 is 𝑛 times 𝑒 to the power of 𝑛𝑑. So, in our case, we just have 𝑛 is equal to π‘Ž. So, we get that 𝑒 prime of 𝑑 is π‘Ž times 𝑒 to the power of π‘Žπ‘‘.

We now need to find an expression for 𝑣 prime of 𝑑. That’s the derivative of the sin of 𝑏𝑑 with respect to 𝑑. And this is a standard trigonometric derivative result. We know for any constant 𝑛, the derivative of the sin of 𝑛𝑑 with respect to 𝑑 is equal to 𝑛 times the cos of 𝑛𝑑. So, using this result with 𝑛 equal to 𝑏, we get that 𝑣 prime of 𝑑 is equal to 𝑏 times the cos of 𝑏𝑑.

We’re now ready to find an expression for our derivative of 𝑓. By the product rule, we have 𝑓 prime of 𝑑 is equal to 𝑣 of 𝑑 times 𝑒 prime of 𝑑 plus 𝑒 of 𝑣 [𝑑] times 𝑣 prime of 𝑑. Substituting in our expressions for 𝑒, 𝑣, 𝑒 prime, and 𝑣 prime, we get that 𝑓 prime of 𝑑 is equal to the sin of 𝑏𝑑 times π‘Žπ‘’ to the power of π‘Žπ‘‘ plus 𝑒 to the power of π‘Žπ‘‘ times 𝑏 cos of 𝑏𝑑. And we can simplify this expression. We will take out our common factor of 𝑒 to the power of π‘Žπ‘‘. Doing this and rearranging, we get 𝑒 to the power of π‘Žπ‘‘ times π‘Ž sin of 𝑏𝑑 plus 𝑏 cos of 𝑏𝑑. And this is our final answer.

Therefore, we’ve shown if 𝑓 of 𝑑 is equal to 𝑒 to the power of π‘Žπ‘‘ times the sin of 𝑏𝑑, then 𝑓 prime of 𝑑 is equal to 𝑒 to the power of π‘Žπ‘‘ times π‘Ž sin of 𝑏𝑑 plus 𝑏 cos of 𝑏𝑑.

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