Differentiation: (lesson 1 of 3)
Common derivatives formulas - exercises
Rules for Common Derivatives
f(x) and g(x) are differentiable functions, C is real number:
1. Constant-Multiple Rule
(f(x)±g(x))′=f′(x)±g′(x)
2. Sum Rule
(C⋅f(x))′=C⋅f′(x)
Derivatives of Polynomials
1. C′=0
Example 1: 15′=0
2. x′=1
3. (xn)′=n⋅xx−1
Example 2: (x5)′=5⋅x5−1=5x4
Example 3: (x51)′=(x−5)′=−5⋅x−5−1=−5x−6=−x65
Example 4: Find the derivative of y=7x4
Solution: (7x4)′=7(x4)′=7⋅4⋅x4−1=28x3
Example 5: Find the derivative of y=2x3−4x2+3x−5
Solution:
(2x3−4x2+3x−5)′=(2x3)′−(4x2)′+(3x)′−5′==2(x3)′−4(x2)′+3x′−0==2⋅3⋅x3−1−4⋅2⋅x2−1+3⋅1==6x2−8x+3
Derivatives of Trigonometric functions
1. (sin(x))′=cos(x)
2. (cos(x))′=−sin(x)
3. (tan(x))′=cos2(x)1=sec2(x)
4. (cot(x))′=−sin2(x)1=−csc2(x)
Example 6:
Find the derivative of y=3x+sin(x)−4cos(x)
Solution 6:
(3x+sin(x)−4cos(x))′=(3x)′+(sin(x))′−(4cos(x))′==3x′+cos(x)−4(cos(x))′==3⋅1+cos(x)−4(cos(x))′==3+cos(x)+4sin(x)
Derivatives of Inverse Trigonometric Functions
1. (arcsin(x))′=1−x21
2. (arccos(x))′=−1−x21
3. (arctan(x))′=1+x21
4. (arccot(x))′=−1+x21
Example 7:
Find the derivative of y=arcsin(x)−2arctan(x)+arcctg(x)
Solution 7:
(arcsin(x)−2arctan(x)+arcctg(x))′=(arcsin(x))′−2(arctan(x))′+(arcctg(x))′==1−x21−21+x21−1+x21==1−x21−31+x21
Derivatives of Exponential and Logarithmic Functions
1. (logax)′=x⋅lna1
2. (lnx)′=x1
3. (ax)′=ax⋅lna
4. (ex)′=ex
Example 8:
Find the derivative of y=3lnx−4ex
Solution 8:
(3lnx−4ex)′=3(lnx)′−4(ex)′=3x1−4ex=x3−4ex