Further Differentiation
Differentiating Exponential and Logarithmic Functions
dy
y = ex ⇒ dx = ex
dy
y = ekx ⇒ dx = kekx
dy
y = ax ⇒ dx = ax ln(a)
dy 1
y = ln(x) ⇒ dx = x
dy f ′ (x)
y = ln[ f (x) ] ⇒ dx = f (x)
Differentiating Trigonometric Functions
dy
y = sin x ⇒ dx = cos x
dy
y = sin kx ⇒ dx = kcos kx
dy
y = cos x ⇒ dx = − sin x
dy
y = cos kx ⇒ dx = − k sin kx
dy 2
y = tan x ⇒ dx = sec x
dy
y = tan kx ⇒ dx = ksec2 kx
Differentiating Functions of ‘y’ with Respect to ‘x’
● Sometimes you want to differentiate ‘y2’ or sin y instead of a function of x
● To do this, follow the golden rule:
u = f (y)
d du dy
dy f (y) = dy × dx
Example
d d dy
dx √y = dy (√y ) × dx
1 dy
= 2√y dx
Differentiating Implicit Functions
● Sometimes x and y are given as a jumbled equation rather than explicitly
saying x= or y=
● In this case, differentiate both sides with respect to x to find the derivative
● You can also apply the product rule to implicit functions
Differentiating Exponential and Logarithmic Functions
dy
y = ex ⇒ dx = ex
dy
y = ekx ⇒ dx = kekx
dy
y = ax ⇒ dx = ax ln(a)
dy 1
y = ln(x) ⇒ dx = x
dy f ′ (x)
y = ln[ f (x) ] ⇒ dx = f (x)
Differentiating Trigonometric Functions
dy
y = sin x ⇒ dx = cos x
dy
y = sin kx ⇒ dx = kcos kx
dy
y = cos x ⇒ dx = − sin x
dy
y = cos kx ⇒ dx = − k sin kx
dy 2
y = tan x ⇒ dx = sec x
dy
y = tan kx ⇒ dx = ksec2 kx
Differentiating Functions of ‘y’ with Respect to ‘x’
● Sometimes you want to differentiate ‘y2’ or sin y instead of a function of x
● To do this, follow the golden rule:
u = f (y)
d du dy
dy f (y) = dy × dx
Example
d d dy
dx √y = dy (√y ) × dx
1 dy
= 2√y dx
Differentiating Implicit Functions
● Sometimes x and y are given as a jumbled equation rather than explicitly
saying x= or y=
● In this case, differentiate both sides with respect to x to find the derivative
● You can also apply the product rule to implicit functions
