Find the derivative of the function y = sin 3x – cos 3x and calculate its value if x = 3П / 4.

| May 11, 2021 | education

Let’s find the derivative of this function: y = sin 3x – cos 3x.

Using the formulas:

(sin x) ’= cos x (derivative of the fundamental elementary function).

(cos x) ’= – sin x (derivative of the fundamental elementary function).

(x ^ n) ’= n * x ^ (n-1) (derivative of the basic elementary function).

(c * u) ’= c * u’, where c is const (basic rule of differentiation).

(u + v) ’= u’ + v ’(basic rule of differentiation).

y = f (g (x)), y ’= f’u (u) * g’x (x), where u = g (x) (basic rule of differentiation).

Thus, the derivative of our function will be as follows:

y ‘= (sin 3x – cos 3x)’ = (sin 3x) ‘- (cos 3x)’ = (3x) ‘* (sin 3x)’ – (3x) ‘* (cos 3x)’ = 3cos 3x – 3 * (- sin 3x) = 3cos 3x + 3sin 3x.

We calculate the value of the derivative at the point x0 = 3π / 4:

y ‘(3π / 4) = 3 * cos (3 * (3π / 4)) + 3 * sin (3 * (3π / 4)) = 3 * cos (9π / 4) + 3 * sin (9π / 4 ) = 3 * cos (2π + (π / 4)) + 3 * sin (2π + (π / 4)) = 3 * cos (π / 4) + 3 * sin (π / 4) = 3 * (cos (π / 4) + sin (π / 4)) = 3 * ((√2 / 2) + (√2 / 2)) = 3 * √2 = 3√2.

Answer: y ‘= 3cos 3x + 3sin 3x, and y’ (3π / 4) = 3√2.



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