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# The derivative of sinx. How to differentiate y = sinx or y = sinf(x)

If you differentiate y = sinx then the derivative is dy/dx = cosx. Moreover, if you differentiate y = sinf(x) then dy/dx = f´(x)cosf(x). Summarising:

**y = sinx → dy/dx = cosx**

**y = sinf(x) → dy/dx = f´(x)cosf(x)**

**Example 1**

Work out the derivate of y = sin2x

Here you have f(x) = 2x and if you differentiate f(x) you get f´(x) = 2

Also the derivative of sin is cos. Therefore using y = sinf(x) → dy/dx = f´(x)cosf(x) you get:

dy/dx = 2cos2x

**Example 2**

Work out the derivate of y = sin(5x-3)

Here you have f(x) = 5x - 3 and if you differentiate f(x) you get f´(x) = 5

Also the derivative of sin is cos. Therefore using y = sinf(x) → dy/dx = f´(x)cosf(x) you get:

dy/dx = 5cos(5x-3)

**Example 3**

Work out the derivate of y = 6sin7x

Here you have f(x) = 7x and if you differentiate f(x) you get f´(x) = 7

Also the derivative of sin is cos. Therefore using y = sinf(x) → dy/dx = f´(x)cosf(x) you get:

dy/dx = 6×7cos7x = 42cos7x

(Note: the 6 in front of 6sin7x can be ignored, but make sure you multiply your final answer by 6)

**Example 4**

Work out the derivate of y = 3sin(4x² + 6x)

Here you have f(x) = 4x² + 6x and if you differentiate f(x) you get f´(x) = 8x + 6

Also the derivative of sin is cos. Therefore using y = sinf(x) → dy/dx = f´(x)cosf(x) you get:

dy/dx = 3×(8x+6) × cos(4x² + 6x) = (24x+18)cos(4x² + 6x)

(Note: the 3 in front of 3sin(4x² + 6x) can be ignored, but make sure you multiply your final answer by 3)

**Example 5**

Work out the derivate of y = -4sin(5x³ + 2x² -3x + 4)

Here you have f(x) = 5x³ + 2x² -3x + 4 and if you differentiate f(x) you get f´(x) = 15x² +4x -3.

Also the derivative of sin is cos. Therefore using y = sinf(x) → dy/dx = f´(x)cosf(x) you get:

dy/dx = -4×(15x² + 4x -3) × cos(5x³ + 2x² -3x + 4) = (-60x² - 16x + 12)cos(5x³ + 2x² -3x + 4)

(Note: like the last 2 examples, the -4 in front of -4sin(5x³ + 2x² -3x + 4) can be ignored, but make sure you multiply your final answer by -4)

So to summarise differentiating sine gives cosine. Harder, example can be carried out using the chain rule or using y = sinf(x) → dy/dx = f´(x)cosf(x).