Explaining The Derivative Of Sin(X) And Its Trigonometric Derivation.

d/dx(sinx)

cosx

The derivative of sin(x) with respect to x is cos(x).

To understand why this is true, we can use the following derivation:

Let’s start with the formula for the derivative of a function f(x) with respect to x:

df/dx = lim (h → 0) (f(x+h) – f(x))/h

Now, let’s substitute sin(x) for f(x):

df/dx = lim (h → 0) (sin(x+h) – sin(x))/h

Using the trigonometric identity sin(a+b) = sin(a)cos(b) + cos(a)sin(b), we can rewrite sin(x+h) as sin(x)cos(h) + cos(x)sin(h):

df/dx = lim (h → 0) [(sin(x)cos(h) + cos(x)sin(h)) – sin(x)]/h

Simplifying the numerator gives:

df/dx = lim (h → 0) (sin(x)cos(h) – sin(x))/h + lim (h → 0) cos(x)sin(h)/h

Factoring sin(x) out of the first limit and using the trigonometric identity lim (h → 0) sin(h)/h = 1, we get:

df/dx = sin(x)lim (h → 0) (cos(h) – 1)/h + cos(x)lim (h → 0) sin(h)/h

Simplifying the limits gives:

df/dx = sin(x)lim (h → 0) (-2sin^2(h/2))/h + cos(x)lim (h → 0) sin(h)/h

Using the trigonometric identity sin^2(h/2) = (1 – cos(h))/2 and simplifying, we get:

df/dx = sin(x)lim (h → 0) -sin(h/2)(1+cos(h))/h + cos(x)lim (h → 0) sin(h)/h

Applying the limit and simplifying further gives:

df/dx = cos(x)

Therefore, the derivative of sin(x) with respect to x is cos(x).

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