Mathematics

Differentiation and Derivatives

55 Questions

Differentiation involves finding the rate of change of functions using established mathematical rules. Topics include the chain rule, quotient rule, and calculating partial derivatives for multivariable equations. Mastery of these mathematical calculations is heavily tested in undergraduate entrance and competitive exams.

Chain rule derivativesQuotient rule applicationPartial derivatives calculationDirectional derivative vectorsLogarithmic function derivatives

Differentiation and Derivatives Questions

Multiple choice maths differencial calculus - differenciability and methods of differnciation differentiation by substitution methods of differentiation derivative of a function

Derivative of ${ \log { x }  }^{ \cos { x }  }$ with respect to $x$ is

  1. ${ \log { x } }^{ \cos { x } }\left[ \cfrac { \cos { x } }{ x\log { x } } -\sin { x } \log { \left( \log { x } \right) } \right] \quad $
  2. ${ \log { x } }^{ \cos { x } }\left[ \cfrac { \cos { x } }{ x\log { x } } -\cos { x } \log { \left( \log { x } \right) } \right] \quad $
  3. ${ \log { x } }^{ \sin { x } }\left[ \cfrac { \sin { x } }{ x\log { x } } -\sin { x } \log { \left( \log { x } \right) } \right] \quad $

  4. None of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Let $y=\log x^{\cos x}$

$\Rightarrow \ y=(\cos x)(\log x)$

$\therefore \dfrac {dy}{dx}=(\cos x) \left (\dfrac {1}{x}\right)+(-\sin x)(\log x)$

$=\dfrac {\cos x}{x}-\sin x\log x$
Multiple choice

What is the derivative of (sin(x)) with respect to (x)?

  1. \(cos(x)\)
  2. \(sec(x)\)
  3. \(-sin(x)\)
  4. \(cosec(x)\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The derivative of (sin(x)) with respect to (x) is (cos(x)).

Multiple choice

What is the exterior derivative of a differential form?

  1. The gradient of the differential form

  2. The divergence of the differential form

  3. The curl of the differential form

  4. The Laplacian of the differential form
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

The exterior derivative of a differential form is a new differential form that measures the boundary of the form.

Multiple choice

What is the purpose of using numerical differentiation in applications?

  1. Estimating the slope of a function at a given point

  2. Finding the roots of polynomials

  3. Solving differential equations

  4. Approximating integrals
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Numerical differentiation is used to approximate the derivative of a function at a given point using numerical methods, which is useful when the analytical derivative is difficult or impossible to obtain.

Multiple choice

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

  1. 3x^2 + 4x - 3

  2. 3x^2 + 4x + 3

  3. 3x^2 - 4x - 3

  4. 3x^2 - 4x + 3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The derivative of a function (f(x)) is given by the limit (\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}). Applying this limit to the function (f(x) = x^3 + 2x^2 - 3x + 4), we get: (\lim_{h \to 0} \frac{(x+h)^3 + 2(x+h)^2 - 3(x+h) + 4 - (x^3 + 2x^2 - 3x + 4)}{h}) = (\lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 + 2x^2 + 4xh + 2h^2 - 3x - 3h + 4 - x^3 - 2x^2 + 3x - 4}{h}) = (\lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3 + 4xh + 2h^2 - 3h}{h}) = (\lim_{h \to 0} \frac{h(3x^2 + 3xh + h^2 + 4x + 2h - 3)}{h}) = (\lim_{h \to 0} (3x^2 + 3xh + h^2 + 4x + 2h - 3)) = (3x^2 + 4x - 3). Therefore, the derivative of the function (f(x) = x^3 + 2x^2 - 3x + 4) is (3x^2 + 4x - 3).

Multiple choice

What is the method of differences?

  1. A method for finding the derivative of a function

  2. A method for finding the integral of a function

  3. A method for finding the limit of a function

  4. A method for finding the power series expansion of a function
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The method of differences is a method for finding the power series expansion of a function.

Multiple choice

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

  1. 3x^2 - 4x + 3

  2. 3x^2 - 2x + 3

  3. 3x^2 - 4x + 1

  4. 3x^2 - 2x + 1
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The derivative of a function f(x) is given by the formula f'(x) = lim_(h->0) [f(x+h) - f(x)] / h. Substituting f(x) = x^3 - 2x^2 + 3x - 4, we get f'(x) = lim_(h->0) [(x+h)^3 - 2(x+h)^2 + 3(x+h) - 4 - (x^3 - 2x^2 + 3x - 4)] / h = lim_(h->0) [x^3 + 3x^2h + 3xh^2 + h^3 - 2x^2 - 4xh - 2h^2 + 3x + 3h - 4 - x^3 + 2x^2 - 3x + 4] / h = lim_(h->0) [3x^2h + 3xh^2 + h^3 - 4xh - 2h^2 + 3h] / h = lim_(h->0) [3x^2 + 3xh + h^2 - 4x - 2h + 3] = 3x^2 - 4x + 3.

Multiple choice

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

  1. 3x^2 - 4x + 3

  2. 3x^2 - 4x + 1

  3. 3x^2 - 2x + 3

  4. 3x^2 - 2x + 1
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The derivative of a function f(x) is given by the formula f'(x) = lim_(h->0) [f(x+h) - f(x)] / h. Substituting the values of f(x), we get f'(x) = lim_(h->0) [(x+h)^3 - 2(x+h)^2 + 3(x+h) - 4 - (x^3 - 2x^2 + 3x - 4)] / h. Simplifying this equation, we get f'(x) = 3x^2 - 4x + 3.

Multiple choice

What is the name of the theorem that states that the derivative of an inverse function is equal to the reciprocal of the derivative of the original function?

  1. Inverse function theorem

  2. Chain rule

  3. Mean value theorem

  4. Cauchy's mean value theorem
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The inverse function theorem states that the derivative of an inverse function is equal to the reciprocal of the derivative of the original function.

Multiple choice

What is the name of the theorem that states that the derivative of a logarithmic function is equal to the reciprocal of the function?

  1. Logarithmic differentiation theorem

  2. Chain rule

  3. Mean value theorem

  4. Cauchy's mean value theorem
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The logarithmic differentiation theorem states that the derivative of a logarithmic function is equal to the reciprocal of the function.

Multiple choice

What is the name of the theorem that states that the derivative of an exponential function is equal to the function itself?

  1. Exponential differentiation theorem

  2. Chain rule

  3. Mean value theorem

  4. Cauchy's mean value theorem
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The exponential differentiation theorem states that the derivative of an exponential function is equal to the function itself.

Multiple choice

The method of approximating the derivative of a function using finite differences is called:

  1. Forward Difference Method

  2. Backward Difference Method

  3. Central Difference Method

  4. Richardson Extrapolation
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Forward Difference Method is a simple numerical method for approximating the derivative of a function.

Multiple choice

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

  1. \(3x^2 - 4x + 3\)
  2. \(3x^2 - 2x + 3\)
  3. \(x^3 - 4x^2 + 3x\)
  4. \(x^3 - 2x^2 + 3\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The derivative of the function (f(x) = x^3 - 2x^2 + 3x - 4) can be found using the power rule of differentiation. The power rule states that if (f(x) = x^n), then (f'(x) = nx^(n-1)). Applying this rule to each term of the function, we get: (f'(x) = 3x^2 - 4x + 3).

Multiple choice

Find the derivative of the function (f(x) = x^3 - 2x^2 + 3x - 4).

  1. \(f'(x) = 3x^2 - 4x + 3\)
  2. \(f'(x) = 3x^2 - 2x + 3\)
  3. \(f'(x) = 3x^2 - 4x - 3\)
  4. \(f'(x) = 3x^2 - 2x - 3\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To find the derivative of the function, we can use the power rule: (\frac{d}{dx}x^n = nx^(n-1)). Applying this rule to each term of the function, we get: (f'(x) = \frac{d}{dx}(x^3 - 2x^2 + 3x - 4) = 3x^2 - 2(2x) + 3(1) - 0 = 3x^2 - 4x + 3).