Differentiation

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What is the geometric meaning of the derivative of a function at a point?

The slope of the normal to the curve at the point

The slope of the curve at the point

The slope of the chord to the curve at the point

The slope of the tangent to the curve at the point (correct)

What is the name of the rule that is used to find the derivative of a function when the limit definition is not applicable?

First principle rule

Chain rule

L'Hopital's rule (correct)

Product rule

What is the application of the derivative of a function at a point?

To find the integral of the function

To find the rate of change of the function at the point (correct)

To find the maximum value of the function

To find the minimum value of the function

What is the notation for the nth derivative of y with respect to x?

D^n y Signup and view all the answers

What is the derivative of (ax+b)^m with respect to x when m>n?

a^n (ax+b)^(m-n) Signup and view all the answers

What is the notation for the value of the nth order derivative at x=a?

f^n(a) Signup and view all the answers

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

a^n sin(ax+b) Signup and view all the answers

What is the notation for the nth successive derivative of y?

D^n y Signup and view all the answers

What is the derivative of the function $f(x) = cos^{-1} (sin x + \frac{x}{2})$ at $x = 1$?

1/2 Signup and view all the answers

If $y = tan^{-1} x$ and $z = sin^{-1} (1 - x^2)$, then what is $\frac{dy}{dx} \div \frac{dz}{dx}$?

-1 Signup and view all the answers

What is the value of the limit $\lim_{x \to \infty} \frac{sin^{-1} x}{x}$?

0 Signup and view all the answers

If $y = tan^{-1} (2x)$, then what is the second derivative of $y$ with respect to $x$?

-\frac{4}{(1 + 4x^2)^2} Signup and view all the answers

If y = cos(x + y), what is the derivative of y with respect to x?

-1 / (1 + x + y) Signup and view all the answers

What is the derivative of y = ln(x + y) with respect to x?

(1 + dy/dx) / (x + y) Signup and view all the answers

If y = tan(x + y), what is the derivative of y with respect to x?

1 / (1 + (x + y)^2) Signup and view all the answers

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

cos(x + y) - y cos(x + y) Signup and view all the answers

If y = e^(x + y), what is the derivative of y with respect to x?

e^(x + y) (1 + dy/dx) Signup and view all the answers

If y = sin⁻¹(x³), then what is y' in terms of x?

³⁻¹ / (1 - x⁶) Signup and view all the answers

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

sinh(x) Signup and view all the answers

If x = sin(θ), then what is dx/dθ in terms of x?

√(1 - x²) Signup and view all the answers

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

sec²(x) Signup and view all the answers

If y = tanh⁻¹(x), then what is y' in terms of x?

1 / (1 - x²) Signup and view all the answers

What is the derivative of $y = sin^{-1} x$ with respect to $x$?

$1 / \sqrt{1 - x^2}$ Signup and view all the answers

If $y = tanh^{-1} x$, then what is the derivative of $y$ with respect to $x$?

$1 / (1 + x^2)$ Signup and view all the answers

What is the derivative of $y = a cos 2x$ with respect to $x$?

$-2a sin 2x$ Signup and view all the answers

What is the derivative of $y = a sin 2x$ with respect to $x$?

$2a cos 2x$ Signup and view all the answers

If $y = \sinh^{-1} x$, then what is $\frac{dy}{dx}$?

$\frac{1}{\sqrt{x^2 + 1}} Signup and view all the answers

If $y = \tan^{-1} \left( \frac{2x}{1 - x^2} \right)$, then what is $\frac{dy}{dx}$?

$\frac{2(1 + x^2)}{(1 - x^2)^2 + 4x^2} Signup and view all the answers

If $y = \sin(x^2)$, then what is $\frac{dy}{dx}$?

$2x \cos(x^2)$ Signup and view all the answers

If $y = \cosh^{-1} (2x)$, then what is $\frac{dy}{dx}$?

$\frac{2}{\sqrt{4x^2 - 1}} Signup and view all the answers