# How do you differentiate # f(x)=(5x^6 - 4) sin(6x) sin(3x) cos(5x) # using the product rule?

f '(x) =

Product rule would be more convenient if trignometrical functions are simplified first. Accordingly,

sin 6x sin 3x cos 5x = (sin 6x cos 5x) sin 3x

By signing up, you agree to our Terms of Service and Privacy Policy

To differentiate ( f(x) = (5x^6 - 4) \sin(6x) \sin(3x) \cos(5x) ) using the product rule, follow these steps:

- Identify the two functions being multiplied: ( u(x) = (5x^6 - 4) \sin(6x) ) and ( v(x) = \sin(3x) \cos(5x) ).
- Apply the product rule: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
- Differentiate ( u(x) ) and ( v(x) ) separately.
- Substitute the derivatives and the original functions into the product rule formula.
- Simplify the expression if necessary.

Let's denote ( u(x) = (5x^6 - 4) \sin(6x) ) and ( v(x) = \sin(3x) \cos(5x) ). Then:

( u'(x) = (30x^5)(\sin(6x)) + (5x^6 - 4)(6\cos(6x)) )

( v'(x) = (3\cos(3x))(\cos(5x)) - (5\sin(3x))(\sin(5x)) )

Substitute these derivatives and original functions into the product rule formula:

( f'(x) = [(30x^5)(\sin(6x)) + (5x^6 - 4)(6\cos(6x))] (\sin(3x) \cos(5x)) + ((5x^6 - 4) \sin(6x)) [(3\cos(3x))(\cos(5x)) - (5\sin(3x))(\sin(5x))] )

This gives the derivative of ( f(x) ) with respect to ( x ).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7