# 19 Lecture

## MTH101

### Midterm & Final Term Short Notes

## Implicit Differentiation

Implicit differentiation is an essential concept in calculus and analytical geometry that helps in finding derivatives of equations that cannot be easily solved for a single variable.

**Important Mcq's**

Midterm & Finalterm Prepration

Past papers included

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**What is the formula for finding the derivative of an implicit function?**

A. dy/dx = f'(x)

B. dx/dy = f'(y)

C. dy/dx = -f'(x)/f'(y)

D. dx/dy = -f'(y)/f'(x)

Answer: C

**What is the first step in implicit differentiation?**

A. Solve for x

B. Solve for y

C. Differentiate both sides with respect to x

D. Differentiate both sides with respect to y

Answer: C

**What is the derivative of y^2 with respect to x using implicit differentiation?**

A. 2y

B. 2xy

C. 2yx

D. 0

Answer: C

**What is the derivative of x^2 + y^2 = 25 with respect to x using implicit differentiation?**

A. dy/dx = -x/y

B. dy/dx = -y/x

C. dy/dx = x/y

D. dy/dx = y/x

Answer: A

**What is the second derivative of y^2 = x^3 using implicit differentiation?**

A. d^2y/dx^2 = -2x/y

B. d^2y/dx^2 = -y/2x

C. d^2y/dx^2 = 2x/y

D. d^2y/dx^2 = y/2x

Answer: B

**What is the derivative of sin(x^2 + y^2) using implicit differentiation?**

A. cos(x^2 + y^2)

B. 2x cos(x^2 + y^2)

C. 2y cos(x^2 + y^2)

D. 2(x+y) cos(x^2 + y^2)

Answer: D

**What is the derivative of y^(1/2) using implicit differentiation?**

A. (1/2) y^(-1/2)

B. (1/2) y^(1/2)

C. (1/2) y^(3/2)

D. (1/2) y^(2)

Answer: A

**What is the derivative of x^2y^3 + xy = 6 using implicit differentiation?**

A. dy/dx = -2x/3y

B. dy/dx = -3y/2x

C. dy/dx = -2y/3x

D. dy/dx = -3x/2y

Answer: C

**What is the equation of the tangent line to x^2 + y^2 = 16 at the point (3, -sqrt(7)) using implicit differentiation?**

A. y = 2x - sqrt(7)

B. y = 2x + sqrt(7)

C. y = -2x - sqrt(7)

D. y = -2x + sqrt(7)

Answer: D

**What is the derivative of ln(xy) using implicit differentiation?**

A. (1/x) + (1/y)

B. (y/x^2) + (x/y^2)

C. (1/y) + (x/y^2)

D. (1/x) + (y/x^2)

Answer: C

**Subjective Short Notes**

Midterm & Finalterm Prepration

Past papers included

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**What is implicit differentiation?**

**Answer:** Implicit differentiation is a method of finding the derivative of an equation that is not in the form of y = f(x) but instead is in the form of an equation that relates x and y.

**Why is implicit differentiation important in calculus and analytical geometry?**

**Answer:** Implicit differentiation is important in calculus and analytical geometry as it helps to find derivatives of equations that cannot be easily solved for a single variable.

**What is the difference between explicit and implicit functions?**

**Answer:** An explicit function is one that can be written as y = f(x), where y is explicitly defined as a function of x. On the other hand, an implicit function is one where the relationship between x and y is not explicitly defined.

**How do you differentiate an implicit function?**

**Answer:** To differentiate an implicit function, you differentiate both sides of the equation with respect to x, treating y as a function of x, and using the chain rule to differentiate any terms that involve y.

**What is the chain rule?**

**Answer:** The chain rule is a rule in calculus that allows you to find the derivative of a composite function.

**Can implicit differentiation be used to find higher-order derivatives?**

**Answer: **Yes, implicit differentiation can be used to find higher-order derivatives of implicit functions.

**How do you find the second derivative using implicit differentiation?**

**Answer: **To find the second derivative using implicit differentiation, you differentiate the first derivative with respect to x.

**Can implicit differentiation be used to find derivatives of equations that are not functions of x and y?**

**Answer: **Yes, implicit differentiation can be used to find derivatives of equations that are not functions of x and y.

**What is the slope of the tangent line to a circle at a given point?**

**Answer: **The slope of the tangent line to a circle at a given point is given by -x/y.

**In which fields is implicit differentiation used?**

**Answer:** Implicit differentiation is used in many fields, including physics, engineering, economics, and other sciences that use calculus.