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To differentiate this, we must use the Chain Rule. This is because we have two functions multiplied by each other:

(x) sin⁡(x).

We use the substitution u = sin(x)....

The first thing to remember is that the tangent to the curve at a given point has the same gradient as the curve at that point.

Let's consider the general case y=f(x) at a point c=(x1,y1)...

Firstly, note that cos^2(x)+5sin^2(x)= cos^2(x) +sin^2(x) +4sin^2(x).

By trignoemtric identies, cos^2(x)+sin^2(x)=1 and so we can just integrate 1+4sin^2(x) since this is equal to cos^2(x...

Using the knowledge of a² = b² + c² - 2bcCos(A)

Rearrange to aquire Cos(A) = (b² + c² - a²)/2bc

Then input ...

d/dx(4x³ + 2x) =  d/dx (4x³) + d/dx(2x)

                       = 4(3x²) + x

                       = 12x² + x