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Differentiation: (lesson 1 of 3)

Common derivatives formulas - exercises

Rules for Common Derivatives

f(x)f(x) and g(x)g(x) are differentiable functions, CC is real number:

1. Constant-Multiple Rule

(f(x)±g(x))=f(x)±g(x)\color{blue}{( f(x) \pm g(x))' = f'(x) \pm g'(x)}

2. Sum Rule

(Cf(x))=Cf(x)\color{blue}{(C \cdot f(x) )' = C \cdot f'(x)}

Derivatives of Polynomials

1. C=0\color{blue}{\text{1. } C' = 0}

Example 1: 15=0 15' = 0

2. x=1\color{blue}{\text{2. } x' = 1}

3. (xn)=nxx1\color{blue}{\text{3. } (x^n)' = n \cdot x^{x-1}}

Example 2: (x5)=5x51=5x4(x^5)' = 5 \cdot x^{5-1} = 5x^4

Example 3: (1x5)=(x5)=5x51=5x6=5x6(\frac{1}{x^5})' = ( x^{-5})' = -5 \cdot x^{-5-1} = -5x^{-6} = -\frac{5}{x^6}

Example 4: Find the derivative of y=7x4y = 7 x^4

Solution: (7x4)=7(x4)=74x41=28x3(7 x^4)' = 7 (x^{\color{red}{4}})' = 7 \cdot \color{red}{4} \cdot x^{\color{red}{4}-1} = 28 x^3

Example 5: Find the derivative of y=2x34x2+3x5y = 2 x^3 - 4 x^2 + 3 x - 5

Solution:

(2x34x2+3x5)=(2x3)(4x2)+(3x)5==2(x3)4(x2)+3x0==23x3142x21+31==6x28x+3 \begin{aligned} (2 x^3 - 4 x^2 + 3x - 5)' &= (2x^3)' - (4x^2)' + (3x)' - 5' = \\ &= 2 (x^{\color{red}{3}})' - 4(x^{\color{red}{2}})' + 3x' - 0 = \\ &= 2 \cdot \color{red}{3} \cdot x^{\color{red}{3} - 1} - 4 \cdot 2 \cdot x^{\color{red}{2} - 1} + 3 \cdot 1 = \\ &= 6 x^2 - 8x + 3 \end{aligned}

Try yourself

(5x312x2+3x14)= \color{blue}{(5x^3 - 12x^2 + 3x - 14)' = }
15x212x6 15x^2 - 12x - 6 15x312x2+6x 15x^3 - 12x^2 + 6x 15x212x+6 15x^2 - 12x + 6 15x224x+3 15x^2 - 24x + 3

Derivatives of Trigonometric functions

1. (sin(x))=cos(x)\color{blue}{\text{1. } (\sin (x))' = \cos (x)}

2. (cos(x))=sin(x)\color{blue}{\text{2. } (\cos (x))' = - \sin (x)}

3. (tan(x))=1cos2(x)=sec2(x)\color{blue}{\text{3. } (\tan (x))' = \frac{1}{\cos^2 (x)} = \sec^2 (x)}

4. (cot(x))=1sin2(x)=csc2(x)\color{blue}{\text{4. } (\cot (x))' = - \frac{1}{\sin^2 (x)} = - \csc^2 (x)}

Example 6:

Find the derivative of y=3x+sin(x)4cos(x)y = 3x + \sin (x) - 4 \cos(x)

Solution 6:

(3x+sin(x)4cos(x))=(3x)+(sin(x))(4cos(x))==3x+cos(x)4(cos(x))==31+cos(x)4(cos(x))==3+cos(x)+4sin(x) \begin{aligned} (3x + \sin (x) - 4 \cos (x))' &= (3x)' + (\sin (x))' - (4 \cos (x))' = \\ &= 3x' + \cos (x) - 4(\cos (x))' = \\ &= 3 \cdot 1 + \cos (x) - 4(\cos(x))' = \\ &= 3 + \cos (x) + 4 \sin (x) \end{aligned}

Try yourself

(x23sin(x)2cos(x)+4)= \color{blue}{(x^2 - 3 \sin (x) - 2 \cos (x) + 4)' = }
x23cos(x)+2sin(x) x^2 - 3 \cos(x) + 2 \sin(x) 2x3cos(x)+2sin(x) 2x - 3 \cos(x) + 2 \sin(x) x+3cos(x)+2sin(x) x + 3 \cos(x) + 2 \sin(x) x+3cos(x)2sin(x) x + 3 \cos(x) - 2 \sin(x)

Derivatives of Inverse Trigonometric Functions

1. (arcsin(x))=11x2\color{blue}{\text{1. } (\arcsin (x))' = \frac{1}{\sqrt{1-x^2}}}

2. (arccos(x))=11x2\color{blue}{\text{2. } (\arccos (x))' = - \frac{1}{\sqrt{1-x^2}}}

3. (arctan(x))=11+x2\color{blue}{\text{3. } (\arctan (x))' = \frac{1}{1+x^2}}

4. (arccot(x))=11+x2\color{blue}{\text{4. } (arccot (x))' = - \frac{1}{1+x^2}}

Example 7:

Find the derivative of y=arcsin(x)2arctan(x)+arcctg(x)y = \arcsin(x) - 2 \arctan(x) + arcctg(x)

Solution 7:

(arcsin(x)2arctan(x)+arcctg(x))=(arcsin(x))2(arctan(x))+(arcctg(x))==11x2211+x211+x2==11x2311+x2 \begin{aligned} (\arcsin (x) - 2 \arctan(x) + arcctg (x))' &= (\arcsin (x))' - 2(\arctan (x))' + (arcctg (x))' = \\ &= \frac{1}{\sqrt{1-x^2}} - 2\frac{1}{1+x^2} - \frac{1}{1+x^2} = \\ &= \frac{1}{\sqrt{1-x^2}} - 3 \frac{1}{1+x^2} \end{aligned}

Try yourself

(x5arctan(x)+arcsin(x))= \color{blue}{(x - 5 \arctan (x) + \arcsin (x))' = }
x2+51+x2+1/sqrt1x2 \frac{x^2 + 5}{1 + x^2} + \frac{1}{/sqrt{1-x^2}} x241+x2+1/sqrt1x2 \frac{x^2 - 4}{1 + x^2} + \frac{1}{/sqrt{1-x^2}} x2+41+x21/sqrt1x2 \frac{x^2 + 4}{1 + x^2} - \frac{1}{/sqrt{1-x^2}} x241+x21/sqrt1x2 \frac{x^2 - 4}{1 + x^2} - \frac{1}{/sqrt{1-x^2}}

Derivatives of Exponential and Logarithmic Functions

1. (logax)=1xlna\color{blue}{\text{1. } (\log_a x)' = \frac{1}{x \cdot \ln a}}

2. (lnx)=1x\color{blue}{\text{2. } (\ln x)' = \frac{1}{x}}

3. (ax)=axlna\color{blue}{\text{3. } (a^x)' = a^x \cdot \ln a}

4. (ex)=ex\color{blue}{\text{4. } (e^x)' = e^x}

Example 8:

Find the derivative of y=3lnx4exy = 3 \ln x - 4e^x

Solution 8:

(3lnx4ex)=3(lnx)4(ex)=31x4ex=3x4ex(3 \ln x - 4e^x)' = 3 (\ln x)' - 4(e^x)' = 3\frac{1}{x} - 4e^x = \frac{3}{x} - 4e^x

Try yourself

(3lnx+x2)= \color{blue}{(3 \ln x + x^2)' = }
2x2+3x \frac{2x^2 + 3}{x} 2x23x \frac{2x^2 - 3}{x} 3x2+2x \frac{3x^2 + 2}{x} 3x22x \frac{3x^2 - 2}{x}