# 20 Lecture

## Derivative of Logarithmic and Exponential Functions

Logarithmic and exponential functions are widely used in mathematics and various other fields of study. In calculus, understanding how to differentiate these functions is crucial for solving complex problems.

## Important Mcq's Midterm & Finalterm Prepration Past papers included

What is the derivative of ln(x)?

a) x

b) 1/x

c) ln(x)

d) 0

Solution: b) 1/x

What is the derivative of e^x?

a) x

b) e^x

c) ln(x)

d) 0

Solution: b) e^x

What is the derivative of ln(u), where u is a function of x?

a) 1/u

b) u/ln(u)

c) u'/ln(u)

d) ln(u)/u'

Solution: c) u'/u

What is the derivative of e^u, where u is a function of x?

a) e^u

b) u'e^u

c) e^(u/x)

d) e^(u^2)

Solution: b) u'e^u

What is the derivative of ln(ax), where a is a constant?

a) 1/xln(a)

b) a/x

c) xln(a)

d) 0

Solution: a) 1/xln(a)

What is the derivative of e^(ax), where a is a constant?

a) ae^x

b) e^(ax)

c) x^a

d) a^x

Solution: a) ae^(ax)

What is the derivative of ln(x^n), where n is a constant?

a) nln(x)

b) n/x

c) x/n

d) 0

Solution: b) n/x

What is the derivative of e^(nx), where n is a constant?

a) e^(nx)

b) n^x

c) ne^(nx)

d) e^(n^x)

Solution: c) ne^(nx)

What is the derivative of ln(e^x)?

a) x

b) 1

c) e^x

d) ln(x)

Solution: b) 1

What is the derivative of e^(ln(x))?

a) x

b) e^x

c) ln(x)

d) 1

Solution: a) x

## Subjective Short Notes Midterm & Finalterm Prepration Past papers included

What is the derivative of ln(x)?

Answer: The derivative of ln(x) is 1/x.

What is the derivative of e^x?

Answer: The derivative of e^x is e^x.

What is the derivative of ln(u), where u is a function of x?

Answer: The derivative of ln(u) is u'/(u).

What is the derivative of e^u, where u is a function of x?

Answer: The derivative of e^u is e^u * u'.

What is the derivative of ln(ax), where a is a constant?

Answer: The derivative of ln(ax) is 1/(xln(a)).

What is the derivative of e^(ax), where a is a constant?

Answer: The derivative of e^(ax) is ae^(ax).

What is the derivative of ln(x^n), where n is a constant?

Answer: The derivative of ln(x^n) is n/x.

What is the derivative of e^(nx), where n is a constant?

Answer: The derivative of e^(nx) is ne^(nx).

What is the derivative of ln(e^x)?

Answer: The derivative of ln(e^x) is 1.

What is the derivative of e^(ln(x))?

Answer: The derivative of e^(ln(x)) is x.

Logarithmic and exponential functions are widely used in mathematics and various other fields of study. In calculus, understanding how to differentiate these functions is crucial for solving complex problems. In this article, we will explore the derivative of logarithmic and exponential functions.

### Derivative of Logarithmic Functions:

The logarithmic function is defined as y = log(x) or ln(x), where x is the argument of the function and y is the output value. The derivative of the natural logarithmic function ln(x) is defined as: dy/dx = 1/x This means that the slope of the tangent line to the natural logarithmic function at any point x is equal to 1/x. The derivative of other logarithmic functions, such as base 10 or base 2 logarithms, can be found using the change of base formula and the chain rule. For example, if we have the function y = log10(x), then: dy/dx = (1/x) * (1/ln(10)) Similarly, for y = log2(x), we have: dy/dx = (1/x) * (1/ln(2))

### Derivative of Exponential Functions:

The exponential function is defined as y = e^x, where e is a mathematical constant approximately equal to 2.71828. The derivative of the exponential function is also an exponential function. dy/dx = d/dx(e^x) = e^x This means that the slope of the tangent line to the exponential function at any point x is equal to e^x. This property is especially useful in solving differential equations involving exponential functions. In addition to the natural exponential function, we can also differentiate other exponential functions using the chain rule. For example, if we have the function y = a^x, where a is a constant, then: dy/dx = d/dx(a^x) = (ln(a)) * (a^x) This formula can be useful in solving problems involving exponential growth or decay, such as population growth or radioactive decay.

### Applications of Logarithmic and Exponential Functions:

Logarithmic and exponential functions have numerous applications in various fields, including finance, physics, and biology. In finance, logarithmic functions are often used to model compound interest and stock prices. Exponential functions are used to model growth or decay of investments or interest rates. In physics, exponential functions are commonly used to describe radioactive decay and electromagnetic waves. Logarithmic functions are used to describe the intensity of sound and light. In biology, exponential functions are used to model population growth and the spread of diseases. Logarithmic functions are used to measure the acidity of solutions and the pH level of the human body. Conclusion: In conclusion, the derivative of logarithmic and exponential functions is an important concept in calculus and has various applications in different fields of study. Understanding these concepts can help in solving complex problems and modeling real-world scenarios.