Find the derivative of each function. y = ( 3 x + 5 ) 2 y=(3x+5)^2 y = ( 3 x + 5 ) 2

In this problem, we're asked to find the derivative of the function y equals 3x plus 5 squared. Now you may be a bit confused here on how exactly we can find this derivative. But remember that whenever a function is being squared, that's the same thing as it multiplying itself. So 3x+5 squared is the same thing as 3x+5 times3x+5. I just have 2 functions multiplying each other, meaning I can use the product rule to find the derivative here. So I have my function on the left, 3x+5. That's the exact same as my function on the right, 3x+5. So let's go ahead and find our derivative here, y prime, using the product rule. So we're gonna take that left function, 3x+5. We're gonna multiply it by the derivative of our right function. We have left d right plus right d left. So taking the derivative of that right hand function, 3x+5, the derivative of this function is just 3, so that's what I'm multiplying by here. Then I'm adding that together with that right hand function, which is the exact same thing, 3x+5 times the derivative of that left hand function, which, again, same function, same derivative. It's just 3. Now from here, we can multiply this out by distributing that 3 into our parentheses in both of these terms here. So y prime is going to be equal to a 9 x plus 15. That's that first term. Don't forget your second term here. It's going to be the exact same thing, 9x+15 yet again. Now from here, we can combine our like terms. We have 9x and 9x and 15 and 15 to give us 18x plus 30 for our final derivative here. Now feel free to let me know if you have any questions, and I'll see you in that next practice problem.

3. Techniques of Differentiation

Product and Quotient Rules

Calculus

3. Techniques of Differentiation

Product and Quotient Rules

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Multiple Choice

Find the derivative of each function.
$y=(3x+5)^2$

$3$

$6x$

$9x+15$

$18x+30$

Verified step by step guidance

Identify the function for which you need to find the derivative: $ y = (3x + 5)^2 $.

Recognize that this is a composite function, which suggests using the chain rule for differentiation. The chain rule states that if you have a function $ y = f(g(x)) $, then the derivative $ y' $ is $ f'(g(x)) \cdot g'(x) $.

Let $ u = 3x + 5 $. Then the function becomes $ y = u^2 $. Differentiate $ y $ with respect to $ u $ to get $ \frac{dy}{du} = 2u $.

Differentiate $ u = 3x + 5 $ with respect to $ x $ to get $ \frac{du}{dx} = 3 $.

Apply the chain rule: $ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 2u \cdot 3 $. Substitute back $ u = 3x + 5 $ to get $ \frac{dy}{dx} = 2(3x + 5) \cdot 3 $. Simplify the expression to find the derivative.