Derivative of Sin(x): Methods and Explained Derivations

derivative sinx

To find the derivative of sin(x), we can use the concept of the derivative of a trigonometric function

To find the derivative of sin(x), we can use the concept of the derivative of a trigonometric function.

The derivative of sin(x) can be found using either the limit definition of the derivative or by using known derivatives of trigonometric functions. I will explain both methods here:

Method 1: Using the limit definition of the derivative:
The derivative of a function is defined as the limit of the difference quotient as the change in x approaches 0. Here’s how we can find the derivative of sin(x) using this definition:

Let f(x) = sin(x). We need to find f'(x), which is the derivative of f(x).

f'(x) = lim(h->0) [f(x+h) – f(x)] / h

= lim(h->0) [sin(x + h) – sin(x)] / h

Using the angle sum formula for sine, sin(A + B) = sin(A)cos(B) + cos(A)sin(B), we can rewrite the above equation as:

f'(x) = lim(h->0) [sin(x)cos(h) + cos(x)sin(h) – sin(x)] / h

= lim(h->0) [sin(x)(cos(h) – 1) + cos(x)sin(h)] / h

Using the identity sin(h)/h -> 1 as h approaches 0, and cos(h) – 1 -> 0 as h approaches 0, we have:

f'(x) = sin(x) * 0 + cos(x) * 1

= cos(x)

So, the derivative of sin(x) is cos(x).

Method 2: Using known derivatives of trigonometric functions:
We can also find the derivative of sin(x) using the known derivatives of trigonometric functions. Here’s how:

The derivative of sin(x) is given by:

d/dx (sin(x)) = cos(x)

This is a well-known result and can be derived using various methods including the definition of derivative or using the unit circle.

In conclusion, the derivative of sin(x) is cos(x) using either the limit definition of the derivative or by using the known derivatives of trigonometric functions.

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