cos^2(x^3)
Find the derivative
How is it -6x^2sin(x^3)cos(x^3)
I know you use chain rule but Im confused what else to apply
the derivative of u^2 is 2u u'
the derivative of cos(u) is -sin(u) u'
so, with u=x^3, the derivative of cos^2(x^3), using the power rule as well, is
2cos(3x^2) * -sin(3x^2) * 6x
Yes, I was able to get that arrangement but I couldnt see how it was the same as -6x^2sin(x^3)cos(x^3)
we could fancy that up even a bit more : -6x^2sin(x^3)cos(x^3)
= (-3x^2)(2sin(x^3)cos(x^3))
= (-3x^2)(sin(2x^3) )
..... by using sin(2A) = 2sinAcosA
wow I did mess that up! The derivative is
2cos(x^3) * -sin(x^3) * 3x^2 = -6x^2 cos(x^3) sin(x^3)
or, as Reiny put it.
To find the derivative of the function cos^2(x^3), you can use the chain rule. The chain rule states that if you have a composition of functions, f(g(x)), where f(u) and g(x) are differentiable functions, then the derivative of f(g(x)) can be found by multiplying the derivative of f(u) with the derivative of g(x), i.e., (f∘g)'(x) = f'(g(x)) * g'(x).
Let's break down the given function:
f(u) = u^2
g(x) = cos(x^3)
To find the derivative of f(g(x)), we need to find the derivative of f(u) and g(x) respectively, and then apply the chain rule.
Step 1: Find the derivative of f(u)
f'(u) = 2u
Step 2: Find the derivative of g(x)
g'(x) = -sin(x^3) * (3x^2)
Applying the chain rule, we differentiate the outer function (cos) and multiply it by the derivative of the inner function (x^3). The derivative of cos(x^3) is -sin(x^3) due to the chain rule, and the derivative of x^3 is 3x^2.
Step 3: Apply the chain rule
Using the chain rule, we can find the derivative of f(g(x)):
(f∘g)'(x) = f'(g(x)) * g'(x)
= 2(cos(x^3))^2 * (-sin(x^3) * (3x^2))
Simplifying further:
= -6x^2 * sin(x^3) * cos(x^3)
Thus, the derivative of cos^2(x^3) is -6x^2 * sin(x^3) * cos(x^3).