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

__Derivatives - Exercises 1__

*Exercise 1.1*

Find derivative of a polynomial

**Σ**

*P(x) =*_{n∈[0,N]}

*A*_{n}·x^{n}*Answer*

*D*

_{x}**Σ**

*P(x) =*_{n∈[1,N]}

*A*_{n}·n·x^{n−1}*Exercise 1.2*

As we know,

*sin(2x)=2sin(x)cos(x)*Find independently derivatives of two functions:

**, as a compound function**

*sin(2x)***, where**

*g(f(x))***and**

*f(x)=2x*

*g(x)=sin(x)*and

**as a product of functions.**

*2sin(x)cos(x)*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})/2Hyperbolic cosine function is defined as

*cosh(x) = (e*^{x}+e^{−x})/2Prove that

*D*and

_{x}**sinh(x) = cosh(x)***D*

_{x}**cosh(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:

**and**

*sec(x) = 1 / cos(x)*

*csc(x) = 1 / sin(x)**Answer*

*D*

_{x}

*sec(x) =*

= sin(x)/cos²(x) =

= sec(x)tan(x)= sin(x)/cos²(x) =

= sec(x)tan(x)

*D*

_{x}

*csc(x) =*

= −cos(x)/sin²(x) =

= −csc(x)cot(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:

**and**

*tan(x) = sin(x) / cos(x)*

*cot(x) = cos(x) / sin(x)**Answer*

There are different variants, all equivalent:

*D*

_{x}

*tan(x) =*

= 1+tan²(x) =

= 1/cos²(x) =

= sec²(x)= 1+tan²(x) =

= 1/cos²(x) =

= sec²(x)

*D*

_{x}

*cot(x) =*

= −1−cot²(x) =

= −1/sin²(x) =

= −csc²(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*

_{x}

*arcsin(x) = (1−x²)*

^{−1/2}*D*_{x}*arccos(x) = −(1−x²)*^{−1/2}
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