# 16 Lecture

## MTH101

### Midterm & Final Term Short Notes

## Techniques of Differentiation

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.

**Important Mcq's**

Midterm & Finalterm Prepration

Past papers included

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**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)

**Subjective Short Notes**

Midterm & Finalterm Prepration

Past papers included

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**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.