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.