differentiate the function
a) y = cos³x
b) y = Sin²xCos3x
c) y = Sin (x³)
look back at your original post of this question
thank you Reiny
a)3cos*2x
To differentiate the given functions, we can use the chain rule, product rule, and power rule of differentiation. Let's differentiate each function separately:
a) y = cos³x:
To differentiate y = cos³x, we can use the power rule of differentiation. The power rule states that if we have a function f(x) raised to the power n, the derivative is given by n * f(x)^(n-1) * f'(x). In this case, f(x) = cos(x), and n = 3:
dy/dx = 3 * cos²x * (-sinx)
= -3sinx * cos²x
b) y = sin²x * cos3x:
To differentiate y = sin²x * cos3x, we will use the product rule of differentiation. The product rule states that if we have two functions f(x) and g(x), the derivative of their product (f(x) * g(x)) is given by f'(x) * g(x) + f(x) * g'(x).
In this case, f(x) = sin²x and g(x) = cos3x. Applying the product rule:
dy/dx = (2sinx * cosx) * cos3x + sin²x * (-3sin3x)
= 2sinx * cosx * cos3x - 3sin³x
c) y = sin(x³):
To differentiate y = sin(x³), we can use the chain rule. The chain rule states that if we have a function f(g(x)), the derivative is given by the derivative of the outer function f'(g(x)) multiplied by the derivative of the inner function g'(x).
In this case, f(x) = sinx and g(x) = x³. Applying the chain rule:
dy/dx = cos(x³) * 3x²
And that's how we differentiate each of the given functions using the appropriate differentiation rules.