Find the derivative of the function.f(x)=sin(3x2)f(x)=\sin(3x^2)

6 x cos ⁡ ( 3 x 2 ) 6x\cos\left(3x^2\right)

Find the derivative of the function. f ( x ) = sin ( 3 x 2 ) f(x)=\sin(3x^2)

this problem, we're asked to find the derivative of f of x equals sine of 3x squared. Now here I see that I have a function inside of another function. So my inside function is that 3x squared. My outside function is the sine of that. So we need to go ahead and use the chain rule here because we have a function in a function. So using the chain rule here to find f prime of x, our derivative here, we're starting from the outside working our way in. So starting from that outside function, sine the derivative of sine is cosine. Remember that whenever we're using the chain rule with trig functions, our argument, since that's our inside function here, that's going to remain as is. So here we're taking the cosine of that inside function unchanged to 3x squared. Then we're accounting for the derivative of that inside function by multiplying by it by it because we're using the chain rule. Now the derivative of 3x squared using the power rule pulling that 2 out to the front, decreasing that power by 1, this becomes 6 x. Now we can rewrite this slightly just to match what we conventionally present these answers as moving that constant and that variable to the front. That is multiplying our trig function cosine of 3x squared. So this is our final derivative here, having used the chain rule. 6x times cosine of 3x squared. Thanks for watching, and I will see you in the next one.

Find the derivative of the function. f ( x ) = sin ⁡ ⁡ ( 3 x 2 ) f(x)=\sin⁡(3x^2)

6 x cos ⁡ ( 3 x 2 ) 6x\cos\left(3x^2\right)

cos ⁡ ( 6 x ) \cos\left(6x\right) cos ( 6 x )

− 6 x cos ⁡ ( 3 x 2 ) -6x\cos\left(3x^2\right) − 6 x cos ( 3 x 2 )

cos ⁡ ( 3 x 2 ) \cos\left(3x^2\right)

3. Techniques of Differentiation

The Chain Rule

Calculus

3. Techniques of Differentiation

The Chain Rule

Struggling with Calculus?

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Multiple Choice

Find the derivative of the function.
$f of x equals sine of open paren 3 x squared close paren$

$\cos\left(6x\right)$

$-6x\cos\left(3x^2\right)$

$6 x the cosine of open paren 3 x squared close paren$

$the cosine of open paren 3 x squared close paren$

Verified step by step guidance

First, identify the function you need to differentiate: f(x) = sin(3x^2) * cos(6x) - 6x * cos(3x^2). This function is a combination of products and differences, so you'll need to apply the product rule and the chain rule.

Apply the product rule to the first term, sin(3x^2) * cos(6x). The product rule states that if you have two functions u(x) and v(x), the derivative of their product is u'(x)v(x) + u(x)v'(x). Here, let u(x) = sin(3x^2) and v(x) = cos(6x).

Differentiate u(x) = sin(3x^2) using the chain rule. The chain rule states that the derivative of sin(g(x)) is cos(g(x)) * g'(x). Here, g(x) = 3x^2, so g'(x) = 6x. Therefore, u'(x) = cos(3x^2) * 6x.

Differentiate v(x) = cos(6x) using the chain rule. The derivative of cos(g(x)) is -sin(g(x)) * g'(x). Here, g(x) = 6x, so g'(x) = 6. Therefore, v'(x) = -sin(6x) * 6.

Now, apply the product rule to the first term: u'(x)v(x) + u(x)v'(x) = (cos(3x^2) * 6x) * cos(6x) + sin(3x^2) * (-sin(6x) * 6). Then, differentiate the second term -6x * cos(3x^2) using the product rule again, and combine all the derivatives to find the derivative of the entire function.

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