How to Find a Derivative

Remember that u=g(x)

Constant Rule-d/dx[C]=0

Power Rule-d/dx[uⁿ]=nu'(u)^n-1

Sin Rule-d/dx[sin(u)]=u'cos(u)

Cos Rule-d/dx[cos(u)]=-u'sin(u)

Tan Rule-d/dx[tan(u)]=u'sec²(u)

Csc Rule-d/dx[csc(u)]=-u'csc(u)cot(u)

Sec Rule-d/dx[sec(u)]=u'sec(u)tan(u)

Cot Rule-d/dx[cot(u)]=-u'csc²(u)

e Rule-d/dx[e^u]=u'e^u

ln Rule-d/dx[ln(u)]=u'/u

a^u Rule-d/dx[a^u]=a^uu'ln(u)

Quotent Rule-d/dx[f(u)/g(u)]=(f'(u)u'g(u)-g'(u)u'f(u))/g(u)²

Product Rule-d/dx[f(u)g(u)]=f'(u)u'g(u)+g'(u)u'f(u)

Chain Rule-d/dx[f(g(x))]=g'(x)f'(g(x))

Example 1

Find the derivative of f(x)=cos(x)

Solution

-sin(x)

Example 2

Find the derivative of f(x)=sin³(2x-3)

Solution

2(3cos²(2x-3))

6cos²(2x-3)

Example 3

Find the derivitive of f(x)=((2x-2)(3x-5))/(3x-4)

Solution

g'(x)=2(3x-5)+3(2x-2)

g'(x)=6x-10+6x-6

g'(x)=12x-16

f'(x)=((12x-16)(3x-4)-3((2x-2)(3x-5))/((3x-4)²))