The derivative of cos and sin. Differentiating sinf(x) and cosf(x).

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The derivative of y = cosf(x) is dy/dx = -f’(x)sinf(x).

The derivative of y = sinf(x) is dy/dx = -f’(x)cosf(x).

Just remember that cosx goes to –sinx and sinx goes to cosx. Let’s take a look at some examples of differentiating cos and sin.

Example 1

Find dy/dx if y = cos(4x-2).

First write down f(x) and differentiate this function to give f'(x) (This goes at the front of your answer)

Here f(x) is 4x-2 so f’(x) is 4. Also the derivative of cos is –sin.

So your answer is dy/dx = -4sin(4x-2)

Example 2

Find dy/dx if y = sin(3x+5).

Again write down f(x) and differentiate this function to give f’(x) (This goes at the front of your answer)

Here f(x) is 3x+5 so f’(x) is 3. Also the derivative of sin is cos.

So your answer is dy/dx = 3cos(3x+5)

Example 3

Find dy/dx if y = cos8x³

Again write down f(x) and differentiate this function to give f’(x) (This goes at the front of your answer)

Here f(x) is 8x³ so f’(x) is 24x². Also the derivative of cos is –sin.

So your answer is dy/dx = -24x²sin(8x³)

Example 4

Find dy/dx if y = sin(9x⁴).

Again write down f(x) and differentiate this function to give f’(x) (This goes at the front of your answer)

Here f(x) is 9x⁴ so f’(x) is 36x³. Also the derivative of sin is cos.

So your answer is dy/dx = 36x³cos(9x⁴)

Summary

Make sure you remember that when you differentiate cosx it goes to negative sinx. Don’t get this mixed up with the derivative of sinx which goes to positive cosx. Don’t get yourself confused – remember the difference between the two.

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