Unizor - Derivatives - Exercises 1

Notes to a video lecture on http://www.unizor.com

Derivatives - Exercises 1

Exercise 1.1
Find derivative of a polynomial
P(x) = Σ_n∈[0,N] A_n·xⁿ
Answer
D_xP(x) = Σ_n∈[1,N] A_n·n·xⁿ⁻¹

Exercise 1.2
As we know,sin(2x)=2sin(x)cos(x)
Find independently derivatives of two functions:
sin(2x), as a compound function g(f(x)), where f(x)=2x and g(x)=sin(x)
and
2sin(x)cos(x) as a product of functions.
Compare the results (supposed to be the same).
Hint
Use an identity
cos(2x)=cos²(x)−sin²(x)

Exercise 1.3
Hyperbolic sine function is defined as
sinh(x) = (e^x−e^−x)/2
Hyperbolic cosine function is defined as
cosh(x) = (e^x+e^−x)/2
Prove that
D_xsinh(x) = cosh(x) and
D_xcosh(x) = sinh(x)
which resembles (except the sign in case of derivative of hyperbolic cosine) the situation with regular sine and cosine.

Exercise 1.4
Find derivative of secant and co-secant using their definitions as reciprocal to cosine and sine:
sec(x) = 1 / cos(x) and
csc(x) = 1 / sin(x)
Answer
D_xsec(x) =
= sin(x)/cos²(x) =
= sec(x)tan(x)
D_xcsc(x) =
= −cos(x)/sin²(x) =
= −csc(x)cot(x)

Exercise 1.5
Find derivative of tangent and cotangent functions using their definitions as ratios of sine and cosine:
tan(x) = sin(x) / cos(x) and
cot(x) = cos(x) / sin(x)
Answer
There are different variants, all equivalent:
D_xtan(x) =
= 1+tan²(x) =
= 1/cos²(x) =
= sec²(x)
D_xcot(x) =
= −1−cot²(x) =
= −1/sin²(x) =
= −csc²(x)

Exercise 1.6
Find derivative of arc-sine and arc-cosine using their definitions as inverse functions to sine and cosine:
φ=arcsin(x) ⇒
⇒ −π/2 ≤ φ ≤ π/2; sin(φ)=x
φ=arccos(x) ⇒
⇒ 0 ≤ φ ≤ π; cos(φ)=x
Answer
D_xarcsin(x) = (1−x²)^−1/2
D_xarccos(x) = −(1−x²)^−1/2