Glossary

To find the derivative of $y = \sin(x^2)$, you'd apply the Chain Rule to get $y' = \cos(x^2) \cdot 2x$.

If $f(x) = 5x^3$, then $f'(x) = 5 \cdot (3x^2) = 15x^2$, applying the Constant Multiple Rule.

When modeling the oscillation of a spring with $x(t) = \cos t$, the velocity of the spring is $v(t) = -\sin t$, found by taking the Derivative of cos x.

If a population grows exponentially according to $P(t) = 100e^t$, then the rate of population growth, $P'(t)$, is also $100e^t$, demonstrating the special property of the Derivative of e^x.

To find the rate at which a quantity modeled by $Q(x) = \ln x$ changes, you'd calculate $Q'(x) = \frac{1}{x}$, using the rule for the Derivative of ln x.

If a particle's position is given by $s(t) = \sin t$, its velocity is $v(t) = \cos t$, meaning its instantaneous rate of change of position is the Derivative of sin x.

To find the rate of change of the volume of a sphere with respect to its radius, $V(r) = \frac{4}{3}\pi r^3$, you'd use the Power Rule to get $V'(r) = 4\pi r^2$.

If you need to find the derivative of $f(x) = x^2 \sin x$, you would use the Product Rule to get $f'(x) = 2x \sin x + x^2 \cos x$.

To differentiate $g(x) = \frac{e^x}{x}$, you'd apply the Quotient Rule to find $g'(x) = \frac{e^x \cdot x - e^x \cdot 1}{x^2}$.

If $s(t)$ is position, $s'(t)$ is velocity, and $s''(t)$ is acceleration, which is the Second Derivative of position.

To find the equation of the Tangent Line to $f(x) = x^2$ at $x=2$, you'd find $f(2)=4$ and $f'(2)=4$, leading to $y-4=4(x-2)$.

Before differentiating $\sin^2 x + \cos^2 x$, you could simplify it to $1$ using a fundamental Trig Identity, making its derivative $0$.