Differentiating sine functions
Transcript of Differentiating sine functions
VCE Maths Methods - Derivatives of circular functions
Differentiating sine functions
1
• The derivative of the sine function is the cosine function.
gradient = 0
gradient = 0
gradient = -1
cosπ =−1
cos 3π
2=0
cosπ
2=0
cos0=1
gradient = 1
y = sin (x)y = cos (x)
VCE Maths Methods - Derivatives of circular functions
Differentiating sine & cosine functions
2
• The derivative of the sin function is the cosine function.
ddx
sin(x )= cos(x )
• The derivative of the cosine function is the (negative) sine function.
ddx
cos(x )=−sin(x )
VCE Maths Methods - Derivatives of circular functions
Differentiating sine functions with different periods
3
• If the sin function has a dilation parallel to the x axis by the factor k, then there will be k periods per 2π and the gradient will be k times greater at the corresponding point along the curve.
y =sin(x ) y =sin(2x )
ddx
sin(2x )=2cos(2x )
VCE Maths Methods - Derivatives of circular functions
Differentiating sine functions with different periods
4
• Use the chain rule to differentiate:
y =sin(kx )y =sin(u),u =kx
ddx
sin(kx )=kcos(kx )
dydx
= dydu
× dudx
= cox(kx )×k
y =sin(kx )
dydu
= cos(u),dudx
=k
ddx
cos(kx )=−ksin(kx )
dydx
= dydu
⋅ dudx
VCE Maths Methods - Derivatives of circular functions
Differentiating tangent functions
5
• The tan function is found from the ratio of sine and cosine functions.
ddx
tan(x )= ddx
sin(x )cos(x )
= 1cos2(x )
=sec2(x ) (Using the quotient rule)
ddx
tanx = 1cos2x
=sec2x
ddx
tankx = kcos2x
=ksec2x (Using the chain rule)
VCE Maths Methods - Derivatives of circular functions
Differentiating tangent functions
6
y =tan(x )
y = 1
cos2(x )
VCE Maths Methods - Derivatives of circular functions
Differentiating circular functions (1)
7
• Find the derivative of the function: f (x )=4sin(2πx )
f '(x )= (4×2π )cos(2πx )
f '(x )=8πcos(2πx )
ddx
sin(kx )=kcos(kx )
VCE Maths Methods - Derivatives of circular functions
Differentiating circular functions (2)
8
y =4cos2(x +π
6)−5• Find the derivative of the function:
y =4cos(u)−5 u =2(x +π
6)=2x +π
3
dydu
=−4sin(u)
dudx
=2
dydx
= dydu
× dudx
=−4sin(u)×2
dydx
=−8sin2(x +π6
)
dydx
= dydu
⋅ dudx
VCE Maths Methods - Derivatives of circular functions
Differentiating circular functions (3)
9
f (x )=−3tan
x2
⎛⎝
⎞⎠ +1• Find the derivative of the function:
f (x )=−3tan
x2
⎛⎝
⎞⎠ +1
f '(x )=− 3
2cos2 x2
⎛⎝
⎞⎠
f '(x )=−12× 3
cos2 x2
⎛⎝
⎞⎠
ddx
tankx = kcos2x