Calculus I — Derivatives & Applications

Derivative

The measure of how a function changes as its input changes; the slope of the tangent line at any point on a function.

Limit definition of a derivative

$f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

Power rule

If $f(x)=x^n$, then $f^{\prime }(x)=n{x}^{n-1}$

Product rule

$(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$

Quotient rule

${\left(\frac{f}{g}\right)}^{\prime }=\frac{f^{\prime }g-f{g}^{\prime }}{g^2}$

Chain rule

If $y=f(g(x))$, then $\frac{dy}{dx}=f^{\prime }(g(x))\cdot g^{\prime }(x)$

What is the derivative of $e^x$?

The derivative of $e^x$ is $e^x$.

Critical point

A point on a function where the derivative is zero or undefined.

Inflection point

A point on a curve where the concavity changes.

Implicit differentiation

Differentiating both sides of an equation with respect to $x$, treating $y$ as a function of $x$.

Second derivative test

A test using the second derivative to determine if a critical point is a local minimum or maximum.

What is the derivative of $\ln (x)$?

The derivative of $\ln (x)$ is $\frac{1}{x}$.

What is the derivative of $\sin (x)$?

The derivative of $\sin (x)$ is $\cos (x)$.

What is the derivative of $\cos (x)$?

The derivative of $\cos (x)$ is $-\sin (x)$.

What is the derivative of $\tan (x)$?

The derivative of $\tan (x)$ is ${\sec }^2(x)$.

Local maximum

A point where a function changes from increasing to decreasing.

Local minimum

A point where a function changes from decreasing to increasing.

Mean Value Theorem

If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, there exists $c\in (a,b)$ such that $f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$.

L'Hôpital's Rule

If ${\lim }_{x\to c}\frac{f(x)}{g(x)}$ is indeterminate of type $\frac{0}{0}$ or $\frac{\infty }{\infty }$, then ${\lim }_{x\to c}\frac{f(x)}{g(x)}={\lim }_{x\to c}\frac{f^{\prime }(x)}{g^{\prime }(x)}$ under certain conditions.

What is the derivative of $\arcsin (x)$?

The derivative of $\arcsin (x)$ is $\frac{1}{\sqrt{1-x^2}}$.

What is the derivative of $\arccos (x)$?

The derivative of $\arccos (x)$ is $-\frac{1}{\sqrt{1-x^2}}$.

What is the derivative of $\arctan (x)$?

The derivative of $\arctan (x)$ is $\frac{1}{1+x^2}$.

Concavity

The property of a curve that describes whether it is curving upwards or downwards.

How do you find the derivative of a composite function?

Use the chain rule: differentiate the outer function, then multiply by the derivative of the inner function.

How do you find the derivative of a function at a point?

Evaluate the derivative function at that point.

How are derivatives applied in optimization problems?

Derivatives find critical points that can indicate local maxima or minima, aiding in optimization.

How do you determine if a function is increasing or decreasing?

Check the sign of the derivative: positive means increasing, negative means decreasing.

Why is the derivative of a constant zero?

A constant function has no rate of change, so its derivative is zero.

What is a tangent line?

A line that touches a curve at a point and has the same slope as the curve at that point.

What does the sign of the second derivative indicate?

A positive second derivative indicates concave up; negative indicates concave down.

What is a higher-order derivative?

A derivative of a derivative, such as the second derivative, third derivative, etc.

What is the derivative of $\frac{1}{x}$?

The derivative of $\frac{1}{x}$ is $-\frac{1}{x^2}$.

What is the derivative of $a^x$ where $a>0$?

The derivative of $a^x$ is $a^x\ln (a)$.

What is the derivative of $x^x$?

The derivative of $x^x$ is $x^x(\ln (x)+1)$.

What is Rolle's Theorem?

A special case of the Mean Value Theorem where $f(a)=f(b)$, implying there exists $c\in (a,b)$ such that $f^{\prime }(c)=0$.

What is a removable discontinuity?

A point at which a function is not defined but could be made continuous by redefining the function at that point.

What are the conditions for using the chain rule?

Both the inner and outer functions must be differentiable.

How does one apply derivatives to graphing?

Derivatives help identify critical points, inflection points, and intervals of increase or decrease.