Calculus I Limits and Derivatives

\(f(x) = 7x^4 - 3x^2 + 5x - 9\text{.}\) Find \(f'(x)\text{.}\)

\(f(x) = (2x^3 + 1)(x^2 - 4)\text{.}\) Find \(f'(x)\) using the Product Rule.

\(f(x) = \frac{5x^2 - 3}{x + 2}\text{.}\) Find \(f'(x)\) using the Quotient Rule.

\(f(x) = (3x^2 - 7)^5\text{.}\) Find \(f'(x)\) using the Chain Rule.

\(f(x) = \sqrt{4x^3 + 1}\text{.}\) Find \(f'(x)\) using the Chain Rule.

\(f(x) = \ln(x^2 + 5)\text{.}\) Find \(f'(x)\text{.}\)

\(f(x) = \ln(\sin x)\text{.}\) Find \(f'(x)\) using the Chain Rule.

\(f(x) = x^3 \cdot \ln(x)\text{.}\) Find \(f'(x)\) using the Product Rule.

\(f(x) = \ln\left(\frac{3x + 1}{x - 2}\right)\text{.}\) Find \(f'(x)\) by using log properties first.

\(f(x) = \log_5(x^2 - 3)\text{.}\) Find \(f'(x)\) using the change-of-base rule.

\(f(x) = e^{4x - 1}\text{.}\) Find \(f'(x)\text{.}\)

\(f(x) = e^{x^2} \cdot x\text{.}\) Find \(f'(x)\) using the Product Rule.

\(f(x) = 3^{2x}\text{.}\) Find \(f'(x)\text{.}\)

\(f(x) = e^{\sin x}\text{.}\) Find \(f'(x)\) using the Chain Rule.

\(f(x) = \sin(3x^2 + 2)\text{.}\) Find \(f'(x)\) using the Chain Rule.

\(f(x) = \cos^3(x)\text{.}\) Find \(f'(x)\) using the Chain Rule.

\(f(x) = \tan(e^x)\text{.}\) Find \(f'(x)\) using the Chain Rule.

\(f(x) = \sin(x) \cdot \cos(x)\text{.}\) Find \(f'(x)\) using the Product Rule.

\(f(x) = \sec(x^2)\text{.}\) Find \(f'(x)\) using the Chain Rule.

\(f(x) = \frac{x^2 \cdot e^x}{\sin(x)}\text{.}\) Find \(f'(x)\) using Quotient and Product Rules.

Find \(\frac{dy}{dx}\) for the equation \(x^2 + y^2 = 25\text{.}\)

Find \(\frac{dy}{dx}\) for \(x^3 + y^3 = 6xy\text{.}\)

Find the derivative \(y'\) of \(\sin(y) = x\text{.}\)

Differentiate implicitly: \(e^y = x^2 + y\text{.}\)

Find \(\frac{dy}{dx}\) for \(xy = 10\text{.}\)

Find \(y'\) for the equation \(x^2 y + y^2 x = 6\text{.}\)

Find \(\frac{dy}{dx}\) at the point (1, 1) for \(x^2 + xy + y^2 = 3\text{.}\)

Differentiate \(\ln(xy) = x + y\text{.}\)

Find \(y'\) for \(\tan(x+y) = x\text{.}\)

Find \(\frac{dy}{dx}\) for \(\sqrt{x} + \sqrt{y} = 4\text{.}\)

Find the slope of the tangent line to the curve \(y = 3x^2 - 5x + 2\) at \(x = 2\text{.}\)

What is the slope of the normal line to the curve \(y = x^3 - 2x\) at the point (1, -1)?

Find the equation of the tangent line to the curve \(y = e^{2x}\) at \(x = 0\text{.}\)

Find the equation of the normal line to the curve \(y = \ln(x)\) at the point (1, 0).

At which point on the curve \(y = x^2 - 4x + 5\) is the tangent line horizontal?

Find the slope of the tangent line to the circle \(x^2 + y^2 = 25\) at the point (3, 4) using implicit differentiation.

What is the equation of the tangent line to the curve \(y = \sin(x)\) at the point (0, 0)?

Find the equation of the normal line to the curve \(y = \frac{4}{x}\) at the point (2, 2).

The curve \(y = ax^2 + bx\) has a tangent line with slope of 5 at the point (1, 3). Find the values of \(a\) and \(b\text{.}\)

Find the \(x\)-coordinate(s) on the curve \(y = \frac{1}{3}x^3 - 2x^2 + 3x\) where the tangent line is parallel to the x-axis.

Find the slope of the tangent line to \(f(x) = e^{3x}\) at \(x = 0\text{.}\)

Find the equation of the normal line to \(f(x) = \ln(x + 1)\) at the point (0, 0).

What is the slope of the tangent line to \(f(x) = \cos(2x)\) at \(x = \pi/4\text{?}\)

Find the equation of the tangent line to \(f(x) = 2^x\) at the point (1, 2).

Find the slope of the normal line to \(f(x) = \tan(x)\) at \(x = \pi/4\text{.}\)

At what value of \(x\) does the function \(f(x) = x e^x\) have a horizontal tangent?

Find the equation of the tangent line to \(f(x) = \ln(x^2)\) at \(x = e\text{.}\)

What is the slope of the normal line to \(f(x) = \sin(x) + \cos(x)\) at \(x = 0\text{?}\)

Find the equation of the tangent line to \(f(x) = e^x \cos(x)\) at (0, 1).

Find the slope of the tangent line to \(f(x) = \log_{10}(x)\) at \(x = 10\text{.}\)