Calculus 1: Derivatives

Constant Multiple Rule

d/dx [cf(x)] = cf'(x)

The Power Rule

d/dx[x^n]=nx^(n-1)

Sum and Difference Rules

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

The Product Rule

d/dx[f(x)g(x)] = f'(x)g(x)+f(x)g'(x)

The Quotient Rule

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

Derivative of sin(x)

cos(x)

Derivative of cos(x)

-sin(x)

Derivative of tan(x)

sec^2(x)

Derivative of csc(x)

-csc(x)cot(x)

Derivative of sec(x)

sec(x)tan(x)

Derivative of cot(x)

-csc^2(x)

The Chain Rule

F'(x) = f'(g(x)) * g'(x)

Derivative of Exponential Functions

d/dx(b^x) = b^x*ln*b

Derivatives of Logs

d/dx(logbx) = 1/(xlnb)

Derivatives of Logs

d/dx(lnx) = 1/x

Derivatives of Logs

d/dx[ln g(x)] = g'(x)/g(x)

Derivatives of Logs

d/dx ln |x| = 1/x

Log Differentation Steps

1. Take the natural log of both sides of an equation y=f(x) and use the Laws of Logarithms to expand the expression

2. Differentiate implicitly with respect to x

3. Solve the resulting equation for y' and replace y by f(x)

cos(y)

sqrt(1-x^2)

Derivative of arcsin

1/sqrt(1-x)

Derivative of arctan

1/1+x^2

Derivative of arccos

- 1/sqrt(1-x^2)

Derivative of arccsc

- 1/xsqrt(x^2 -1)

Derivative of arcsec

1/xsqrt(x^2 -1)

Derivative of arccot

- 1/(1+x^2)