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

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

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

this problem, we're asked to find the derivative of f of x equals sine of three x squared. Now here I see that I have a function inside of another function. So my inside function is that three x 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 three x 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 three x squared using the power rule pulling that two out to the front, decreasing that power by one, this becomes a six 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 three x squared. So this is our final derivative here, having used the chain rule. Six x times cosine of three x squared. Thanks for watching, and I'll see you in the next one.

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

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

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) cos ( 3 x 2 )

12. Trigonometric Functions

Derivatives of Trig Functions

Business Calculus

12. Trigonometric Functions

Derivatives of Trig Functions

Struggling with Business Calculus?

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Multiple Choice

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

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

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

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

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

Verified step by step guidance

Step 1: Identify the function to differentiate. The given function is a product of multiple terms: f(x) = sin(3x^2) * cos(6x) - 6x * cos(3x^2). This requires the use of the product rule and chain rule for differentiation.

Step 2: Apply the product rule to the first term, sin(3x^2) * cos(6x). The product rule states that if h(x) = u(x) * v(x), then h'(x) = u'(x) * v(x) + u(x) * v'(x). Here, u(x) = sin(3x^2) and v(x) = cos(6x).

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

Step 4: Differentiate v(x) = cos(6x). The derivative of cos(g(x)) is -sin(g(x)) * g'(x). Here, g(x) = 6x, so g'(x) = 6. Thus, v'(x) = -6 * sin(6x).

Step 5: Combine the results using the product rule for the first term and differentiate the second term, -6x * cos(3x^2), using the product rule again. Simplify the expression to find the derivative of the entire function.

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