Function Form | Derivative |
---|
Power Rule: y=cxn+C | y′=cnxn−1 |
Sum/Difference Rule: y=u(x)±v(x) | y′=u′(x)±v′(x) |
Product Rule: y=u(x)⋅v(x) | y′=u′(x)v(x)+u(x)v′(x) |
Quotient Rule: y=v(x)u(x) | y′=v2(x)u′(x)v(x)−u(x)v′(x) |
General Power Rule: y=u(x)v(x) | y′=uv(uu′v+v′lnu) |
Function | Derivative |
---|
y=ex | y′=ex |
y=ekx | y′=kekx |
y=ax | y′=axlna |
y=eu(x) | y′=u′(x)eu(x) (Chain Rule) |
Function | Derivative |
---|
y=lnx | y′=x1 |
y=logax | y′=xlna1 |
y=lnu(x) | y′=u(x)u′(x) (Logarithmic Differentiation) |
Function | Derivative |
---|
y=sinx | y′=cosx |
y=cosx | y′=−sinx |
y=tanx | y′=sec2x=1+tan2x |
y=cotx | y′=−csc2x=−(1+cot2x) |
y=secx | y′=secxtanx |
y=cscx | y′=−cscxcotx |
Function | Derivative |
---|
y=arcsinx | y′=1−x21 |
y=arccosx | y′=−1−x21 |
y=arctanx | y′=1+x21 |
y=arccotx | y′=−1+x21 |
Function | Derivative |
---|
y=sinhx | y′=coshx |
y=coshx | y′=sinhx |
y=tanhx | y′=sech2x |
y=cothx | y′=−csch2x |
If y=f(u(x)), then:
dxdy=dudy⋅dxdu=f′(u)⋅u′(x)
Given x=f(t),y=g(t):
dxdy=x˙y˙=f′(t)g′(t)
Second derivative:
dx2d2y=x˙3x˙y¨−y˙x¨
For equations where y is not explicitly expressed in terms of x, differentiate both sides and solve for y′.
Example:
For x2+y2=r2,
2x+2ydxdy=0⟹dxdy=−yx
If y=f(x) has an inverse x=f−1(y), then:
dxdy=dydx1
Example:
For y=arccosx,
x=cosy⟹dydx=−siny⟹dxdy=−1−x21