Do You Know Your Derivative Rules?

1) What does the equation (d/dx) .(c^ax)= c^ax (lnc .a), c>0 mean?

That the equation above is true for all c, but the derivative for c=0 yields a complex number

That the equation above is true for all c, but the derivative for c>0 yields a complex number

That the equation above is true for all c, but the derivative for c

That the equation above is true for all c, but the derivative for c yields a complex number.

not-available-via-ai

Explanation

not-available-via-ai

2) What is the ideal number sentence describing logarithmic derivative?

Lnf= f/f

(lnf)'=f'/f

Lnf'=f/f'

Lnf'=f/1

The ideal number sentence describing the logarithmic derivative is (lnf)'=f'/f. This equation represents the derivative of the natural logarithm of a function f, which is equal to the derivative of f divided by f itself.

Explanation

The ideal number sentence describing the logarithmic derivative is (lnf)'=f'/f. This equation represents the derivative of the natural logarithm of a function f, which is equal to the derivative of f divided by f itself.

3) What are the 2 arguments that define an inverse tangent?

F'(x,y)

(y,x)

Arctan (y,x).

F(x)'

The correct answer is "arctan (y,x)." The inverse tangent function, arctan, takes two arguments, y and x, and returns the angle whose tangent is the ratio of y to x. In other words, it gives the angle whose tangent is y/x. Therefore, arctan (y,x) is the correct expression for the inverse tangent with two arguments.

Explanation

The correct answer is "arctan (y,x)." The inverse tangent function, arctan, takes two arguments, y and x, and returns the angle whose tangent is the ratio of y to x. In other words, it gives the angle whose tangent is y/x. Therefore, arctan (y,x) is the correct expression for the inverse tangent with two arguments.

4) What is the number sentence for the digamma function?

ψ0(x)

0(x)

ψ(x)

ψ0(x) +1

The number sentence for the digamma function is represented by ψ0(x). This function is also known as the psi function or the logarithmic derivative of the gamma function. It is used in various mathematical and statistical calculations, particularly in the field of number theory. The notation ψ0(x) specifically represents the value of the digamma function at the given input x.

Explanation

The number sentence for the digamma function is represented by ψ0(x). This function is also known as the psi function or the logarithmic derivative of the gamma function. It is used in various mathematical and statistical calculations, particularly in the field of number theory. The notation ψ0(x) specifically represents the value of the digamma function at the given input x.

5) What is the set of rules that can help compute the derivative function?

Fa di Bruno's theorem and the general Leibniz rule.

Faa di Bruno's formula and the general Leibniz rule.

Faa di Bruno's theorem and the general Leibniz theorem.

The Bruno's theorem and the Leibnis rule.

The correct answer is Faa di Bruno's formula and the general Leibniz rule. Faa di Bruno's formula is a mathematical formula that provides a way to compute higher-order derivatives of composite functions. The general Leibniz rule, on the other hand, is a rule that allows us to compute the derivative of a product of two functions. Together, these two rules provide a comprehensive set of rules for computing the derivative function.

Explanation

The correct answer is Faa di Bruno's formula and the general Leibniz rule. Faa di Bruno's formula is a mathematical formula that provides a way to compute higher-order derivatives of composite functions. The general Leibniz rule, on the other hand, is a rule that allows us to compute the derivative of a product of two functions. Together, these two rules provide a comprehensive set of rules for computing the derivative function.

6) What are the central objects of study in a complex analysis?

The polymorphic function

The holomorphic function

The morphological rule

The isomorphic rule

The central objects of study in complex analysis are holomorphic functions. Holomorphic functions are complex-valued functions that are differentiable at every point within their domain. They are important in complex analysis because they have many properties that make them analytically tractable, such as the Cauchy-Riemann equations. By studying holomorphic functions, mathematicians can understand and analyze complex functions and their behavior, leading to applications in various areas of mathematics and physics.

Explanation

The central objects of study in complex analysis are holomorphic functions. Holomorphic functions are complex-valued functions that are differentiable at every point within their domain. They are important in complex analysis because they have many properties that make them analytically tractable, such as the Cauchy-Riemann equations. By studying holomorphic functions, mathematicians can understand and analyze complex functions and their behavior, leading to applications in various areas of mathematics and physics.

7) What is the Schwarzian derivative?

It is one that describes how a function is approximated by a fractional-linear map.

It is one that describes how a complex function is approximated by a fractional-linear map.

It is one that describes how a simple function is approximated by a fractional-linear map.

It is one that describes how a compounded function is approximated by a fractional-linear map.

The Schwarzian derivative is a mathematical concept that describes how a complex function is approximated by a fractional-linear map. This derivative is used in complex analysis to study the behavior of functions and their mappings.

Explanation

The Schwarzian derivative is a mathematical concept that describes how a complex function is approximated by a fractional-linear map. This derivative is used in complex analysis to study the behavior of functions and their mappings.

8) Define the Wirtinger derivative? (pick the right answer.)

It's a set of differential operators permitting the construction of a differential calculus for complex functions.

It is one that completes a complex function is approximated by a fractional-linear map.

It is one that describes how a simple function is approximated by a fractional-linear map.

It is one that describes how a ordinary function is approximated by a fractional-linear map.

The Wirtinger derivative is a set of differential operators that allow for the development of a differential calculus specifically designed for complex functions. These operators are used to differentiate complex functions with respect to their complex variables, providing a framework for analyzing and manipulating complex-valued functions.

Explanation

The Wirtinger derivative is a set of differential operators that allow for the development of a differential calculus specifically designed for complex functions. These operators are used to differentiate complex functions with respect to their complex variables, providing a framework for analyzing and manipulating complex-valued functions.

9) What is the role of the functional derivative?

It is one with respect to a function of a functional on a space of formulas.

It is one with respect to a function of a functional on a space of functions.

It is one with respect to a function of a functional on a space of rules.

It is one with respect to a function of a functional on a space of domains.

The role of the functional derivative is to find the rate of change of a functional with respect to a function. In this context, the functional derivative is taken with respect to a function of a functional on a space of functions. This means that the functional derivative allows us to determine how the value of a functional changes as the function it depends on varies.

Explanation

The role of the functional derivative is to find the rate of change of a functional with respect to a function. In this context, the functional derivative is taken with respect to a function of a functional on a space of functions. This means that the functional derivative allows us to determine how the value of a functional changes as the function it depends on varies.

10) Explain the Frechet derivative?

It allows the extension of the directions derivatives to a general Banach space.

It allows the restrictions of the directions derivatives to a general Banach space.

It allows the construction of the directions derivatives to a general Banach space.

It shrinks the extension of the directions derivatives to a general Banach space.

The Frechet derivative is a concept in mathematics that allows for the extension of directional derivatives to a general Banach space. This means that it provides a way to calculate derivatives in a more general setting, beyond just Euclidean spaces. By allowing for this extension, the Frechet derivative enables the study and analysis of functions in more abstract spaces, which is important in many areas of mathematics and physics.

Explanation

The Frechet derivative is a concept in mathematics that allows for the extension of directional derivatives to a general Banach space. This means that it provides a way to calculate derivatives in a more general setting, beyond just Euclidean spaces. By allowing for this extension, the Frechet derivative enables the study and analysis of functions in more abstract spaces, which is important in many areas of mathematics and physics.