differentiate each function
a) y = cos^3x
b) y = sin(x^3)
c) y = sin^2 xcos3x
my answers:
a) y' = 3cos^2 x(-sinx)
b) y' = cos(x^3)(3x^2)
c) for this one, i don't know which one is correct. i got 2 different answers but i think they may be the same.. im not sure, i did it twice and got 2 diff answers, please let me know which is right or if they are both wrong. first answer for c: y' = 2sinxcosxcos3x + 3cos2x(-sinx)sin^2x or 2nd answer for c: y' = 2sinxcosxcos3x + (-sin3x)(3)(sin^2x)
thank you
c) y = sin^2 xcos3x
y' = sin^2 x (-3 sin 3x) + cos 3x (2 sin x cos x)
That is your second answer. Your first answer has a cos 2x in it which comes from where I do not know.
thank you
To differentiate each function, you can follow these steps:
a) y = cos^3x
To differentiate this function, you can use the chain rule. First, we differentiate the outer function (cos^3x) with respect to the inner function (x). The derivative of cos^3x is 3cos^2x(-sinx).
So the correct answer is: y' = 3cos^2x(-sinx)
b) y = sin(x^3)
To differentiate this function, we again apply the chain rule. First, we differentiate the outer function (sin(x^3)) with respect to the inner function (x^3). The derivative of sin(x^3) is cos(x^3)(3x^2).
So the correct answer is: y' = cos(x^3)(3x^2)
c) y = sin^2xcos3x
To differentiate this function, we can use the product rule. The product rule states that if you have two functions u(x) and v(x), then the derivative of their product u(x)v(x) is given by u'(x)v(x) + u(x)v'(x).
Let's break down the function y = sin^2xcos3x into two parts: u(x) = sin^2x and v(x) = cos3x.
Now let's find the derivatives of each part:
- The derivative of sin^2x is 2sin(x)cos(x) (using the chain rule).
- The derivative of cos3x is -sin(3x)(3) (using the chain rule and the derivative of cosx).
Now apply the product rule:
y' = u'(x)v(x) + u(x)v'(x)
= (2sin(x)cos(x))cos3x + sin^2x(-sin(3x)(3))
So both of your answers are correct:
- First answer for c: y' = 2sinxcosxcos3x + 3cos2x(-sinx)sin^2x
- Second answer for c: y' = 2sinxcosxcos3x + (-sin3x)(3)(sin^2x)
Both expressions represent the derivative of y = sin^2xcos3x.