Correct Answer - Option 4 : 2 sin(4x + 10)
Concept:
Derivative of sinx with respect to x is cosx
Chain rule:
Let y = f(v) be a differentiable function of v and v = g(x) be a differentiable function of x then \(\frac{{dy}}{{dx}} = \frac{{dy}}{{dv}} ⋅ \frac{{dv}}{{dx}}\)
Calculation:
Given function is y = sin2(2x+5)
We differentiate the function with respect to x
⇒ y' = [sin2(2x+5)]'
As we know that, \(\frac{{dy}}{{dx}} = \frac{{dy}}{{dv}} ⋅ \frac{{dv}}{{dx}}\)
⇒ y' = 2 sin(2x+5) ⋅ [sin(2x+5)]'
⇒ y' = 2 sin(2x+5) ⋅ cos(2x+5) ⋅ (2x+5)'
⇒ y' = 2 sin(2x+5).cos(2x+5).(2)
⇒ y' = 2 sin[2(2x+5)] (∴ sin2x = 2sinx.cosx)
⇒ y' = 2 sin(4x+10)
Hence, option 4 is correct.
