Derivative Rules and Definitions (3.2-3.6)

Calc 1

Product Rule

d/dx (f(x)g(x)) = f(x)g’(x) + g(x)f’(x)

Find the derivative of y = (x-7)(2x+1)

f(x) = x-7 g(x) = 2x+1

f’(x) = 1 g’(x) = 2

y’ = (x-7)(2) + (2x+1)(1)

Find the derivative of h(x) = 7x²e^x

f(x) = 7x² g(x) = e^x

f’(x) = 14x g’(x) = e^x

h’(x) = (7x²)(e^x) + (e^x)(14x)

Quotient Rule

d/dx (f(x)/g(x)) = g(x)f’(x) - f(x)g’(x) / [g(x)]²

*ORDER IS IMPORTANT

Find the derivative of y = x²+x-2 / x³+6

f(x) = x²+x-2 g(x) = x³+6

f’(x) + 2x+1 g’(x) = 3x²

y’ = (x³+6)(2x+1) - (x²+x-2)(3x²) / [x³+6]²

Find the derivative of h(x) = e^x+x / 1-x²

f(x) = e^x+x g(x) = 1-x²

f’(x) + e^x+1 g’(x) = -2x

h’(x) = (1-x²)(e^x+1) - (e^x+x)(-2x) / [1-x²]²

d/dx [sin(x)]

cos x

d/dx [cos(x)]

-sin(x)

Differentiate y = e^xsin(x)

f(x) = e^x g(x) = sin(x)

f’(x) + e^x g’(x) = cos(x)

y’ = (e^x)(cos(x))+(sin(x))(e^x)

d/dx [tan(x)]

sec²x

d/dx [sec(x)]

sec(x)tan(x)

d/dx [cot(x)]

-csc²x

d/dx [csc(x)]

-csc(x)cot(x)

Chain Rule

d/dx [f(g(x))] = f’(g(x))g’(x)

dy/dx = (dy/du)(du/dx)

d/dx [log_b x]

1/xlnb

d/dx [ln x]

1/x