Learn how to find the derivative of sin(x) using a formula – Step-by-step guide with examples

deriv of sinx

cosx

The derivative of sin(x) can be found using the following formula:

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

This means that the derivative of sin(x) with respect to x is equal to cos(x).

To understand this formula, we can use the definition of the derivative. The derivative of a function represents the rate of change of that function at a given point.

For sin(x), the rate of change at any point is given by the cosine function. This is because the derivative measures the change in the function as we move along the x-axis, and the cosine function represents the change in the slope of the sine curve.

So, if we take the derivative of sin(x), we get:

d/dx sin(x) = 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 this as:

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

Simplifying this expression by cancelling out sin(x) terms, we get:

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

Now, taking the limit as h approaches 0 using L’Hopital’s Rule, we get:

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

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

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