16 Lecture

What is the derivative of f(x) = x^3 + 4x^2 - 5x - 2?

a) f'(x) = 3x^2 + 8x - 5

b) f'(x) = 3x^2 + 8x + 5

c) f'(x) = 3x^3 + 8x^2 - 5x - 2

d) f'(x) = 3x^2 + 4x - 5

Solution: The derivative of f(x) is f'(x) = 3x^2 + 8x - 5. Therefore, the correct answer is an option (a).

What is the derivative of f(x) = sin(x)cos(x)?

a) f'(x) = cos(x)sin(x)

b) f'(x) = cos^2(x) - sin^2(x)

c) f'(x) = -sin(x)cos(x)

d) f'(x) = 2cos(x)sin(x)

Solution: Using the product rule, we get f'(x) = cos(x)cos(x) - sin(x)sin(x) = cos^2(x) - sin^2(x). Therefore, the correct answer is option (b).

What is the derivative of f(x) = 3x^4 - 2x^3 + 5x^2 - 4x + 1?

a) f'(x) = 12x^3 - 6x^2 + 10x - 4

b) f'(x) = 12x^3 - 6x^2 + 5x - 4

c) f'(x) = 3x^3 - 2x^2 + 5x - 4

d) f'(x) = 3x^3 - 2x^2 + 10x - 4

Solution: The derivative of f(x) is f'(x) = 12x^3 - 6x^2 + 10x - 4. Therefore, the correct answer is option (a).

What is the derivative of f(x) = e^x cos(x)?

a) f'(x) = e^x sin(x)

b) f'(x) = e^x(cos(x) + sin(x))

c) f'(x) = e^x(cos(x) - sin(x))

d) f'(x) = e^x(cos(x) - cos(x))

Solution: Using the product rule, we get f'(x) = e^x cos(x) - e^x sin(x) = e^x(cos(x) - sin(x)). Therefore, the correct answer is option (c).

What is the derivative of f(x) = ln(5x)?

a) f'(x) = 1/(5x)

b) f'(x) = 5ln(x)

c) f'(x) = 5/(ln(x))

d) f'(x) = 1/x

Solution: Using the chain rule, we get f'(x) = 1/(5x). Therefore, the correct answer is option (a).

What is the derivative of f(x) = x^2 ln(x)?

a) f'(x) = 2x ln(x) + x

b) f'(x) = x ln(x)

c) f'(x) = 2x ln(x) + 2x

d) f'(x) = 2x ln(x) + x^2

Solution: Using (a)

What is the power rule of differentiation?

Answer: The power rule states that the derivative of a function of the form f(x) = x^n is given by f'(x) = nx^(n-1).

How is the product rule used to find the derivative of a product of two functions?

Answer: The product rule states that if f(x) and g(x) are two functions, then the derivative of their product is given by the formula f'(x)g(x) + f(x)g'(x).

What is the chain rule used for in differentiation?

Answer: The chain rule is used to find the derivative of a composite function.

How is the quotient rule used to find the derivative of a quotient of two functions?

Answer: The quotient rule states that if f(x) and g(x) are two functions, then the derivative of their quotient f(x)/g(x) is given by the formula (f'(x)g(x) - f(x)g'(x))/(g(x))^2.

How are trigonometric identities used to simplify the derivatives of trigonometric functions?

Answer: Trigonometric identities can be used to simplify the derivatives of trigonometric functions and make them easier to compute.

What is logarithmic differentiation used for?

Answer: Logarithmic differentiation is a technique used to find the derivative of a function that is difficult to differentiate using other methods.

How is implicit differentiation used to find the derivative of an implicitly defined function?

Answer: Implicit differentiation is used to find the derivative of a function that is defined implicitly by an equation.

What is the difference between explicit and implicit differentiation?

Answer: Explicit differentiation is used to find the derivative of a function that is defined explicitly in terms of its independent variable, while implicit differentiation is used to find the derivative of a function that is defined implicitly by an equation.

What is the derivative of a constant function?

Answer: The derivative of a constant function is 0.

What is the derivative of the natural logarithm function?

Answer: The derivative of the natural logarithm function f(x) = ln(x) is given by f'(x) = 1/x.

Calculus is a branch of mathematics that deals with the study of change and motion. One of the fundamental concepts in calculus is differentiation, which involves finding the rate at which a function changes. The derivative of a function at a particular point is the slope of the tangent line to the function at that point. In this article, we will explore various techniques of differentiation.

Power Rule

The power rule is one of the most basic and frequently used rules of differentiation. According to this rule, the derivative of a function of the form f(x) = x^n is given by f'(x) = nx^(n-1). For example, the derivative of f(x) = x^3 is f'(x) = 3x^2.

Product Rule

The product rule is used to find the derivative of a product of two functions. According to this rule, if f(x) and g(x) are two functions, then the derivative of their product is given by the formula f'(x)g(x) + f(x)g'(x). For example, if f(x) = x^2 and g(x) = sin(x), then the derivative of f(x)g(x) is f'(x)g(x) + f(x)g'(x) = 2x sin(x) + x^2 cos(x).

Chain Rule

The chain rule is used to find the derivative of a composite function. According to this rule, if f(x) and g(x) are two functions, then the derivative of their composite function f(g(x)) is given by the formula f'(g(x))g'(x). For example, if f(x) = sin(x) and g(x) = x^2, then the derivative of f(g(x)) is f'(g(x))g'(x) = cos(x^2)2x.

Quotient Rule

The quotient rule is used to find the derivative of a quotient of two functions. According to this rule, if f(x) and g(x) are two functions, then the derivative of their quotient f(x)/g(x) is given by the formula (f'(x)g(x) - f(x)g'(x))/(g(x))^2. For example, if f(x) = x^2 and g(x) = cos(x), then the derivative of f(x)/g(x) is (f'(x)g(x) - f(x)g'(x))/(g(x))^2 = (2x cos(x) + x^2 sin(x))/(cos(x))^2.

Trigonometric Identities

Trigonometric identities are used to simplify the derivatives of trigonometric functions. Some of the most commonly used trigonometric identities include sin^2(x) + cos^2(x) = 1, sin(x+y) = sin(x)cos(y) + cos(x)sin(y), and cos(x+y) = cos(x)cos(y) - sin(x)sin(y). These identities can be used to simplify the derivatives of trigonometric functions and make them easier to compute.

Logarithmic Differentiation

Logarithmic differentiation is a technique used to find the derivative of a function that is difficult to differentiate using other methods. According to this technique, if f(x) is a function, then its derivative can be expressed as (ln(f(x)))'. For example, if f(x) = x^x, then its derivative can be found using logarithmic differentiation as follows: (ln(x^x))' = (x ln(x))'/x = ln(x) + 1.

Implicit Differentiation

Implicit differentiation is used to find the derivative of a function that is defined implicitly by an equation.