Find the derivative of the given function.y=x2e3x2+5xy=x^2e^{3x^2+5x}y=x2e3x2+5x

x e 3 x 2 + 5 ( 6 x 2 + 5 x + 2 ) xe^{3x^2+5}\left(6x^2+5x+2\right)

Find the derivative of the given function. y = x 2 e 3 x 2 + 5 x y=x^2e^{3x^2+5x} y = x 2 e 3 x 2 + 5 x

In this problem, we're asked to find the derivative of the function y equals x squared times e to the power of 3 x squared plus 5 x. Now here, we're going to have to use both the product rule and the chain rule. If you do not yet know those rules for finding derivatives, go ahead and circle back to this problem after you've learned those rules. For the rest of you, let's get into this one. We wanna find our derivative here, dydx, using the product rule since we have x squared multiplying e to the power of 3x squared plus 5x, remembering our memory tool for the product rule, left d right plus a right d left. Our left hand function is, in this case, x squared. So x squared times the derivative of e to the power of 3x squared plus 5x. So this is going to be e to the power of 3x squared plus 5x. And then using the chain rule here, multiplying by the derivative of what's in that exponent, The derivative of 3x squared plus 5x is going to be 6x plus 5. So that's half of our product rule. Then we're adding this together with a right d left. So e to the power of 3x squared plus 5x, that's our function that's on the right, multiplied by the derivative of x squared, which is, in this case, 2x. Now we can go ahead and factor something out here because each of these terms have an x in common, and they also have an e to the power of 3x squared+ 5x in common. So I can go ahead and pull that out. That's x times e to the power of 3x squared plus 5x. And then this is multiplied by x times 6x+5 plus 2. And we can take this a little bit further because we can simplify this much more. X times 6x is going to be 6x squared and then plus 5x plus 2. So rewriting this one final time, this is x times e to the 3x squared plus 5. And then inside of those parentheses, we have 6x squared plus 5x plus 2, and this is our final derivative here, dydx. Let me know if you have any questions, and let's keep going.

Find the derivative of the given function. y = x 2 e 3 x 2 + 5 x y=x^2e^{3x^2+5x} y = x 2 e 3 x 2 + 5 x

x e 3 x 2 + 5 ( 6 x 2 + 5 x + 2 ) xe^{3x^2+5}\left(6x^2+5x+2\right)

2 x e 3 x 2 + 5 x − 1 2xe^{3x^2+5x-1} 2 x e 3 x 2 + 5 x − 1

2 x e 3 x 2 + 5 x + x 2 e 3 x 2 + 5 x 2xe^{3x^2+5x}+x^2e^{3x^2+5x} 2 x e 3 x 2 + 5 x + x 2 e 3 x 2 + 5 x

2 x e 3 x 2 + 5 x − 1 ( 6 x + 5 ) 2xe^{3x^2+5x-1}(6x+5) 2 x e 3 x 2 + 5 x − 1 ( 6 x + 5 )

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

Calculus

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

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Multiple Choice

Find the derivative of the given function.
$y=x^2e^{3x^2+5x}$

$2xe^{3x^2+5x-1}$

$2xe^{3x^2+5x}+x^2e^{3x^2+5x}$

$2xe^{3x^2+5x-1}(6x+5)$

$x e^{3 x^{2} + 5} (6 x^{2} + 5 x + 2)$

Verified step by step guidance

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

Recognize that this is a product of two functions: $ u(x) = x^2 $ and $ v(x) = e^{3x^2 + 5x} $. Use the product rule for differentiation, which states that $ (uv)' = u'v + uv' $.

Differentiate $ u(x) = x^2 $ to get $ u'(x) = 2x $.

Differentiate $ v(x) = e^{3x^2 + 5x} $ using the chain rule. First, differentiate the exponent $ 3x^2 + 5x $ to get $ 6x + 5 $, then multiply by the original function $ e^{3x^2 + 5x} $. Thus, $ v'(x) = (6x + 5)e^{3x^2 + 5x} $.

Apply the product rule: $ y' = u'v + uv' = 2x e^{3x^2 + 5x} + x^2 (6x + 5)e^{3x^2 + 5x} $. Simplify the expression to get the final derivative.

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