Derivative of Function

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1. Derivative of a Power Function

• Function:

$$f(x)=x^{3}$$

• Derivative:

$$f\acute{}(x)=3x^{2}$$

• 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:

$$f(x)=2x^{4}+3x^{2}-5$$

• Derivative:

$$f\acute{}(x)=8x^{3}+6x$$

• 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:

$$f(x)=7x^{5}$$

• Derivative:

$$f\acute{}(x)=35x^{4}$$

• Python Code:

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

• Output:

35*x**4

4. Derivative of a Sum of Functions

• Function:

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

• Derivative:

$$f\acute{}(x)=2x+1$$

• Python Code:

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

5. Derivative of a Trigonometric Function

• Function:

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

• Derivative:

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

• 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:

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

• Derivative (Product Rule):

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

• Python Code:

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

• Output:

x*cos(x) + sin(x)

7. Derivative of a Quotient

• Function:

$$f(x)=\frac{x^{2}+1}{x}$$

• Derivative (Quotient Rule):

$$f\acute{}(x)=\frac{2x(x)-(1)(x^{2}+1)}{x^{2}}=\frac{x^{2}-1}{x^{2}}$$

• 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:

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

• Derivative (Chain Rule):

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

• 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:

$$f(x)=e^{x}$$

• Derivative:

$$f\acute{}(x)=e^{x}$$

• 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:

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

• Derivative:

$$f\acute{}(x)=\frac{1}{x}$$

• Python Code:

from sympy import ln

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

• Output:

1/x

Keywords

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