Question

D/dx sin(4x+2)??

Answer 1

`4cos(4x+2)`

Explanation:

`d/(dx) sin(4x+2) = 4cos(4x+2)`

Recall:
`d/(dx) sin x = cos x`

Answer 2

`4cos(4x+2)`

Explanation:

We use the chain rule, which states that,

`dy/dx=dy/(du)*(du)/dx`

Let `y=sin(4x+2)`, so we need to find `dy/dx`.

We let `u=4x+2,:du=4dx,(du)/dx=4`.

Then, `y=sin(u),dy=cos(u)du,dy/(du)=cos(u)`.

Combining our results together, we find that:

`dy/dx=cos(u)*4`

`=4cos(u)`

Reversing back our substitution that we made earlier, we get the final answer:

`color(blue)(=4cos(4x+2))`

Answer 3

`4cos(4x+2)`

Explanation:

We have a composite function here, so we know we'll have to use the Chain Rule:

If we have a composite function `f(g(x))`, the derivative will be

`f'(g(x))*g'(x)`

In our function `sin(4x+2)`, we know:

`f(x)=sinx=>color(red)(f'(x)=cosx)`

`color(blue)(g(x)=4x+2)=>color(lime)(g'(x)=4)`

Now, we can input into the Chain Rule. We get:

`color(lime)4*color(red)(cos(color(blue)(4x+2))`

`=>4cos(4x+2)`

Chain Rule Shortcut*: `d/dx sin(bx+c)= bcos(bx+c)`

*For sine of a linear function

When we're taking the derivative of the sine of a linear function, we just have to realize:

• The coefficient of `x` will come out front
• `sin` is replaced with `cos` (because `d/dx sinx=cosx`)
• The inner function remains

Hope this helps!