The Derivative of Sin(x) | Step-by-Step Calculation and Simplification

(d/dx) sin(x)

To find the derivative of sin(x), let’s use the definition of the derivative:

The derivative of a function f(x) with respect to x, or (d/dx)f(x), is defined as the limit as h approaches 0 of the difference quotient:

(d/dx)f(x) = lim(h→0) [(f(x+h) – f(x))/h]

Now, let’s apply this definition to find the derivative of sin(x):

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

To simplify this expression, we can use the trigonometric identity known as the difference of sine formula:

sin(A) – sin(B) = 2 * cos((A + B)/2) * sin((A – B)/2)

Applying this formula to our expression, we have:

(d/dx)sin(x) = lim(h→0) [2 * cos((x + h + x)/2) * sin((x + h – x)/2)/h]

Now, we can simplify further

To find the derivative of sin(x), let’s use the definition of the derivative:

The derivative of a function f(x) with respect to x, or (d/dx)f(x), is defined as the limit as h approaches 0 of the difference quotient:

(d/dx)f(x) = lim(h→0) [(f(x+h) – f(x))/h]

Now, let’s apply this definition to find the derivative of sin(x):

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

To simplify this expression, we can use the trigonometric identity known as the difference of sine formula:

sin(A) – sin(B) = 2 * cos((A + B)/2) * sin((A – B)/2)

Applying this formula to our expression, we have:

(d/dx)sin(x) = lim(h→0) [2 * cos((x + h + x)/2) * sin((x + h – x)/2)/h]

Now, we can simplify further. The cos((x + h + x)/2) can be written as cos((2x + h)/2), which simplifies to cos(x + h/2). Similarly, the sin((x + h – x)/2) simplifies to sin(h/2).

(d/dx)sin(x) = lim(h→0) [2 * cos(x + h/2) * sin(h/2)/h]

Now, let’s take the limit as h approaches 0. As h approaches 0, both cos(x + h/2) and sin(h/2) approach 1.

(d/dx)sin(x) = lim(h→0) [2 * 1 * 1/h]

Simplifying further, we have:

(d/dx)sin(x) = lim(h→0) [2/h]

As h approaches 0, the limit becomes infinite. Therefore, the derivative of sin(x) is:

(d/dx)sin(x) = 2

So, the derivative of sin(x) with respect to x is simply 2.

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