Objective Questions

What is the derivative of \( y = 3x^2 + 2 \) with respect to \( x \)?

Find \( \frac{dy}{dx} \) if \( y = \sin x \).

If \( f(x) = \frac{x^2 - 9}{x + 3} \) has a removable discontinuity at \( x = -…

Evaluate \( \lim_{x \to \infty} \frac{10^8 x^5 + 10^6 x^4 + 10^4 x^2}{10^9 x^6 …

If \( f(x) = \sqrt{4\sin 2x + 2} \), find \( f'(0) \).

Let \( f \) and \( g \) be differentiable functions with \( f(1)=4, g(1)=3, f'(…

What is \( \lim_{h \to 0} \frac{\cos(\frac{\pi}{2} + h) - \cos(\frac{\pi}{2})}{…

If \( f(x) = \sin^2(3-x) \), find \( f'(0) \).

If \( \lim_{x \to 2} \frac{f(x)}{x-2} = f'(2) = 0 \), which must be true?

If \( f(x) = x-1 \) and \( g(x) = x^2+1 \), then \( f(g(x)) = g(f(x)) \) when \…

What is the horizontal asymptote of \( f(x) = \frac{(x-14)^2}{(x+18)(x-14)} \)?

What is the equation of the tangent line to the curve \( x^2 + y^2 = 169 \) at …

A point moves along \( y = x^2 + 1 \). When \( x = 4 \), the \( x \)-coordinate…

The tangent to \( y = \frac{kx+8}{k+x} \) at \( x = -2 \) is \( y = x + 4 \). F…

Find \( f'(x) \) for \( f(x) = \frac{x}{\sqrt{x^2 - 4}} \).

If \( \frac{dW}{dt} = \frac{1}{25}(W - 300) \), find \( \frac{d^2W}{dt^2} \) in…

Find the fourth derivative of \( h(t) = 3t^7 - 6t^4 + 8t^3 - 12t + 18 \).

What is the second derivative of \( g(x) = \sin(2x^3 - 9x) \)?

Find \( \frac{dy}{dx} \) if \( 3x^2 + 3y^2 = 2 \).

Differentiate \( y = x^6 - 7x^2 \) with respect to \( x \).

Find the gradient of the curve \( y = x^2 - 9x + 13 \) at \( x = 6 \).

Find the stationary points of \( y = x^3 - 3x^2 - 9x + 3 \).

Determine the nature of the stationary point at \( x = 3 \) for \( y = x^3 - 3x…

Find the range of \( x \) for which \( y = x^3 - 6x^2 + 12 \) is decreasing.

Differentiate \( y = x^{2x} \).

Find the derivative of \( y = \sqrt{x-1} \) at \( x=5 \).

What is the derivative of \( y = e^{3x} \)?

Find \( dy/dx \) if \( y = \ln(x^2 + 1) \).

A boat leaves harbor A heading East at 15 mph. Boat B heads South at 20 mph. Si…

If \( y = x \ln x \), what is the derivative?

Find the derivative of \( y = \tan(2x) \).

What is the derivative of \( y = 5e^x / \cos x \)?

Find \( dy/dx \) for \( x = 2t^2, y = 4t \).

Derivative of \( y = \sin^{-1} x \) is?

What is the derivative of a constant?

Find \( dy/dx \) for \( y = (x^2+1)/(x^2-1) \).

Find \( dy/dx \) if \( y = \ln(\sin x) \).

What is the derivative of \( y = e^{\tan x} \)?

Evaluate \( d/dx(x^2 \sin x) \).

Derivative of \( y = 10x^2 \) is?

What is the derivative of \( y = \ln 7 \)?

Find \( dy/dx \) for \( y = (x+1)/x \).

A cylinder's radius increases at 2 cm/sec, height decreases at 3 cm/sec. Rate o…

Find \( dy/dx \) for \( y = \log_7(2x - 3) \).

Find \( dy/dx \) for \( y = \sin(x^2) \).

What is the derivative of \( y = e^{-5z} + 8\ln(2z^4) \) with respect to \( z \…

Find \( f'(x) \) for \( f(x) = ax^2 + bx + c \).

Stationary points of \( y = 2x^3 - 6x^2 + 12 \) are?

If \( h(t) = (at+b)^2 \), find the rate of change of \( N \) with respect to \(…

Find \( dy/dx \) if \( 5y^2 = 2x^3 - 5y \).

What is the slope of the curve \( y = x^4 + x + 1 \) at \( x=1 \)?

What is the derivative of \( y = 6\sin x \)?

Find \( d/dt(4t^2 - 7t + 5) \).

What is the derivative of \( y = 1/\sqrt{x} \)?

Stationary points of \( y = x^2 + 16/x \) are?

Differentiate \( y = e^{-x} \).

Derivative of \( y = x^6 \cos x \) is?

Find \( dy/dx \) for \( y = \frac{x}{x+1} \).

Find the equation of the normal to \( y = x^3 - 4x^2 + 1 \) at \( x=2 \).

Differentiate \( y = 1/x \).

Find \( dy/dx \) if \( 5x^3 = -3xy + 2 \).

What is the derivative of \( y = 3x^2 + 3 \)?

If \( h(x) = f(x) + g(x) \), find \( h'(x) \).

Find the derivative of \( y = x^5 \).

Evaluate \( d/dz(1/4 z^2 - 1/z) \).

Find the gradient at \( x=3 \) for \( y = x^3 - 11x + 1 \).

Find \( dy/dx \) for \( y = 7x + \sqrt{x} \).

If \( y = (2x-1)(4x+3) \), find \( dy/dx \).

Differentiate \( y = 2x - 8/x^2 \).

Find the derivative of \( y = \sqrt{1 + \sqrt{x}} \).

Stationary points of \( y = 2x^2 + 1/x \) are?

Find \( dy/dx \) if \( xy + \sin y = 1 \).

A soap bubble volume decreases at 6 cm^3/sec. Formula \( V = (p - qt)^2 \). At …

Find the second derivative of \( y = \ln(7 - x^3) \).

Differentiate \( y = e^{x \ln x} \).

Find \( dy/dx \) if \( y = (x^2+2x+1)(x^3+3x) \).

If \( y = x^5 \sin 3x \tan 5x \), find \( dy/dx \).

What is the derivative of \( y = \frac{1}{x^2+4} \)?

If \( P = at^2 - bt \), rate of change of \( P \) with respect to \( t \) is?

Value of \( k \) for the function \( y = (4x\sqrt{x} + k)/7x \) if tangent at \…

Find \( f'(0) \) for \( f(x) = 1 - 2\sin x \).

Evaluate the fourth derivative of \( x^3 - x^2 + x - 1 \).

Find the second derivative of \( Q(v) = 2(6 + 2v - v^2)^4 \).

Find the coordinate of \( Q \) for \( y = 2x^3 - 9x^2 + 12x - 10 \) where gradi…

Find \( a, b \) if \( (2, 2) \) is on \( y = ax^3 + bx^2 - 10 \) and gradient t…

Stationary point of \( y = x(x^2 - 128\sqrt{x}) \) is at?

If \( V = 2 - 2\pi x^3 \), find \( dV/dx \).

Differentiate \( y = (x^5 + 7x^2 + 2)^7 \).

When evaluating the limit of \( f(x) = \frac{7 + x}{9 - x} \) as \( x \) approa…

Which of the following is considered an indeterminate form that requires furthe…

What is the limit of \( x^2 + 3 \) as \( x \) tends to 2?

Why can the limit of \( \frac{x(x+1)}{x} \) as \( x \) tends to 0 not be evalua…

According to the properties of limits, the limit of a sum \( [f(x) + g(x)] \) i…

Evaluate the limit of \( \frac{x^3 - 8}{x^2 - 4} \) as \( x \) tends to 2.

What is the result of dividing any non-zero number by infinity \( \left( \frac{…

In the rationalisation method for limits involving square roots, what is the co…

Evaluate the limit of \( \frac{\sqrt{x} - 1}{x - 1} \) as \( x \) tends to 1.

Evaluate the limit of \( \frac{x - 1}{\sqrt{x^2 + 3} - 2} \) as \( x \) tends t…

When evaluating a limit as \( x \) approaches infinity for a rational function,…

Evaluate the limit of \( \frac{2x^4 - 3x^2 + 1}{6x^4 + x^3 - 3x} \) as \( x \) …