Question

Question #5bad2

Answer 1

`5cos5x - 7sin7x`

Explanation:

`d/dx sincolor(red)(5x) + coscolor(blue)(7x)`

Use the Chain Rule, differentiate outside, then differentiate inside

`color(red)((f(g))' = f'(g)*g')`

To break it down,

First, differentiate `sin(... )`,

`d/dx sin( ...) = cos (... )`

I am using sin( ... ) to show that we are differentiating the outer part first, and ignore the original `5x` first.

Secondly, differentiate inside the parenthesis.

`d/dx ...color(red)(5x) = 5`

Multiply cos(...) with 5.

The proper working is as such:

`d/dx sin5x = 5* cos 5x`

Apply the same workings to `cos 7x`.

`d/dx cos(...) = -sin(...)`
`d/dx ...7x = 7`

Multiply -sin(...) with 7

Thus, `d/dx cos7x = -7sin7x`

`color(blue)(d/dx (sin5x + cos7x) = 5cos5x - 7 sin7x)`