Chapter 3 Differentiation Rules Flashcards
Derivative of a constant function?
d/dx(c) = 0
Power rule?
d/dx (x^n) = nx^n-1
Contant Multiple Rule?
d/dx [c f(x)] = c d/dx f(x)
Sum Rule?
d/dx [f(x) + g(x)] = d/dx f(x) + d/dx g(x)
Difference Rule?
d/dx [f(x) - g(x)] = d/dx f(x) -d/dx g(x)
Derivative of Natural Exponential Function?
d/dx e^x = e^x
Product Rule?
d/dx [f(x) g(x)] = f(x) d/dx [g(x)] + g(x) d/dx [f(x)]
Quotient Rule?
d/dx [f(x)/g(x)] = [ g(x) d/x [f(x)] - f(x) d/dx [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
Chain Rule
If g is differentiable at x and f is differentiable at g(x), then the composite function F = f g defined by F’(x) = f’(g(x)) * g’(x)
Power Rule Combined With Chain Rule
d/dx[g(x)]^n = n[g(x)]^n-1 * g’(x)
Derivative of General Exponential Functions
d/dx (b^x) = b^x ln b
Implicit Differentiation
We don’t solve an equation for y in terms of x in order to find the derivative of y. Instead, you differentiate both sides of the equation with respect to x and then solve the result dy/dx equation.
Derivative of a Logarithmic Function
d/dx (log b x) = 1 / x ln b
Steps in logarithmic differentiation
- Take natural logs of both sides of an equation and use the Laws of Logarithms to expand equation
- Differentiate implicity with respect to X
- Solve the resulting equation for y’ and replace y by f(x)
d/dx sin^-1 x
1/(1-x^2)^1/2
d/dx cos^-1 x
- [1/(1-x^2)^1/2]
d/dx tan^-1 x
1/ (1+x^2)
d/dx csc^-1 x
- [1/x *(x^2+1)^1/2]
d/dx cot^-1 x
-[1 / (1+x^2)]
