Python Encyclopedia for Academics
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  • 1. Derivative of a Power Function
  • 2. Derivative of a Polynomial
  • 3. Derivative of a Constant Multiplied Function
  • 4. Derivative of a Sum of Functions
  • 5. Derivative of a Trigonometric Function
  • 6. Derivative of a Product of Two Functions
  • 7. Derivative of a Quotient
  • 8. Derivative of a Composite Function
  • 9. Derivative of an Exponential Function
  • 10. Derivative of a Logarithmic Function
  • Keywords
  1. Mathematics
  2. Calculus 1

Derivative of Function

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1. Derivative of a Power Function

  • Function:

f(x)=x3f(x)=x^{3}f(x)=x3

  • Derivative:

fˊ(x)=3x2f\acute{}(x)=3x^{2}fˊ(x)=3x2

  • Python Code:

from sympy import symbols, diff

x = symbols('x')
f = x**3
derivative = diff(f, x)
print(derivative)

  • Output:

3*x**2

2. Derivative of a Polynomial

  • Function:

f(x)=2x4+3x2−5f(x)=2x^{4}+3x^{2}-5f(x)=2x4+3x2−5

  • Derivative:

fˊ(x)=8x3+6xf\acute{}(x)=8x^{3}+6xfˊ(x)=8x3+6x

  • Python Code:

f = 2*x**4 + 3*x**2 - 5
derivative = diff(f, x)
print(derivative)

  • Output:

8*x**3 + 6*x

3. Derivative of a Constant Multiplied Function

  • Function:

f(x)=7x5f(x)=7x^{5}f(x)=7x5

  • Derivative:

fˊ(x)=35x4f\acute{}(x)=35x^{4}fˊ(x)=35x4

  • Python Code:

f = 7*x**5
derivative = diff(f, x)
print(derivative)

  • Output:

35*x**4

4. Derivative of a Sum of Functions

  • Function:

f(x)=x2+x+1f(x)=x^{2}+x+1f(x)=x2+x+1

  • Derivative:

fˊ(x)=2x+1f\acute{}(x)=2x+1fˊ(x)=2x+1

  • Python Code:

f = x**2 + x + 1
derivative = diff(f, x)
print(derivative)

5. Derivative of a Trigonometric Function

  • Function:

f(x)=sin(x)f(x)=sin(x)f(x)=sin(x)

  • Derivative:

fˊ(x)=cos(x)f\acute{}(x)=cos(x)fˊ(x)=cos(x)

  • Python Code:

from sympy import sin, cos

f = sin(x)
derivative = diff(f, x)
print(derivative)

  • Output:

cos(x)

6. Derivative of a Product of Two Functions

  • Function:

f(x)=x.sin(x)f(x)=x.sin(x)f(x)=x.sin(x)

  • Derivative (Product Rule):

fˊ(x)=x.cos(x)+sin(x)f\acute{}(x)=x.cos(x)+sin(x)fˊ(x)=x.cos(x)+sin(x)

  • Python Code:

f = x * sin(x)
derivative = diff(f, x)
print(derivative)

  • Output:

x*cos(x) + sin(x)

7. Derivative of a Quotient

  • Function:

f(x)=x2+1xf(x)=\frac{x^{2}+1}{x}f(x)=xx2+1​

  • Derivative (Quotient Rule):

fˊ(x)=2x(x)−(1)(x2+1)x2=x2−1x2f\acute{}(x)=\frac{2x(x)-(1)(x^{2}+1)}{x^{2}}=\frac{x^{2}-1}{x^{2}}fˊ(x)=x22x(x)−(1)(x2+1)​=x2x2−1​

  • Python Code:

f = (x**2 + 1) / x
derivative = diff(f, x)
print(derivative)

  • Output:

2 - (x**2 + 1)/x**2

8. Derivative of a Composite Function

  • Function:

f(x)=sin(x2)f(x)=sin(x^{2})f(x)=sin(x2)

  • Derivative (Chain Rule):

fˊ(x)=(2x).cos(x2)f\acute{}(x)=(2x).cos(x^{2})fˊ(x)=(2x).cos(x2)

  • Python Code:

f = sin(x**2)
derivative = diff(f, x)
print(derivative)

  • Output:

2*x*cos(x**2)

9. Derivative of an Exponential Function

  • Function:

f(x)=exf(x)=e^{x}f(x)=ex

  • Derivative:

fˊ(x)=exf\acute{}(x)=e^{x}fˊ(x)=ex

  • Python Code:

from sympy import exp

f = exp(x)
derivative = diff(f, x)
print(derivative)

  • Output:

exp(x)

10. Derivative of a Logarithmic Function

  • Function:

f(x)=ln(x)f(x)=ln(x)f(x)=ln(x)

  • Derivative:

fˊ(x)=1xf\acute{}(x)=\frac{1}{x}fˊ(x)=x1​

  • Python Code:

from sympy import ln

f = ln(x)
derivative = diff(f, x)
print(derivative)

  • Output:

1/x

Keywords

derivative,calculus,calculus 1,python,sympy,math,mathematics,differentiation,power rule,product rule,quotient rule,chain rule,polynomial,trigonometric function,exponential function,logarithmic function,symbolic computation,automatic differentiation,step-by-step,python code, nerd cafe

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