Calculus (MATH 1500): Differentiations

What’s the constant rule?

d/dx (C) = 0

What’s the power rule?

d/dx (xⁿ) = nx^n-1

What’s the constant multiple rule?

d/dx [C(f(x))] = (C) d/dx f(x)

What’s the definition of a derivative?

$$f^{\prime}\left(x\right)=\lim_{h\to0}\frac{f\left(x+h\right)-f\left(x\right)}{h}$$

What’s the product rule?

\frac{d}{\differentialD x}\left\lbrack f\left(x\right)\cdot g\left(x\right)\right\rbrack=f^{\prime}\cdot g+f\cdot g^{\prime}

What’s the quotient rule?

\frac{d}{\differentialD x}\left(\frac{f}{g}\right)=\frac{gf^{\prime}-fg^{\prime}}{g^2}

What’s the chain rule?

\frac{d}{\differentialD x}\left\lbrack f\left(g\left(u\right)\right)\right\rbrack=f^{\prime}\left(g\left(u\right)\right)\cdot g^{\prime}\left(u\right)\cdot u^{;^{\prime}}

How do you differentiate exponential functions featuring euler’s number?

\frac{d}{\differentialD x}\left(e^{u}\right)=e^{u}\cdot u^{\prime}

lne = 1

How do you differentiate exponential functions featuring a constant?

\frac{d}{\differentialD x}\left\lbrack a^{u}\right\rbrack=a^{u}\cdot u^{^{\prime}}\cdot\ln a

How do you differentiate natural logarithms?

\frac{d}{\differentialD x}\left\lbrack\ln\left(u\right)\right\rbrack=\frac{u^{\prime}}{u}

How do you differentiate logarithms?

\frac{d}{\differentialD x}\left\lbrack\log_{a}\left(u\right)\right\rbrack=\frac{u^{\prime}}{u\cdot\ln a}

How do you differentiate sinx, tanx, and secx?

\frac{d}{\differentialD x}\left(\sin x\right)=\cos x

\frac{d}{\differentialD x}\left(\tan x\right)=\sec^2x

\frac{d}{\differentialD x}\left(\sec x\right)=\sec x\tan x

How do you differentiate cosx, cotx, and cscx?

\frac{d}{\differentialD x}\left\lbrack\cos\left(x\right)\right\rbrack=-\sin\left(x\right)

\frac{d}{\differentialD x}\cot x=-\csc^2x

\frac{d}{\differentialD x}\left\lbrack\csc\left(x\right)\right\rbrack=-\csc\left(x\right)\cot\left(x\right)

How do you implicitly differentiate?

> Differentiate both sides wrt x but whenever differentiating a y-term: always include dy/dx.

> Collect all dy/dx terms on one side.

> Factor out dy/dx.

> Divide to solve for dy/dx.

How do you differentiate stuff like x^x?

1. Set y = x^x.

2. Set lny = lnx^x.

3. Move: lny = xlnx

4. Differentiate both sides

5. Sub y for x^x

How do you use the definition to find the tangent line at the point a indicated?

> Use derivative definition to find f’(a).

> Find the point (a, f(a)).

> Use the point slope form y - f(a) = f’(a)(x-a).

How do you use the definition to (a) find the derivative and (b) the tangent line at a point?

> Use the derivative definition to find f’(x).

> Using the found f’(x), evaluate at said point a to get slope.

> Find the point (a, f(a)).

> Use the point slope form y - f(a) = f’(a)(x-a).

How do you find constants so that the piecewise function is differentiable?

> Identify all breakpoint(s) where the formula changes → apply continuity at each breakpoint → obtain equations involving the constants. (LHL = f(a) = RHL)

> Differentiate each piece → apply differentiability at each break point → obtain additional equation(s). (LHD = RHD)

> Solve the resulting system for unknown constants.

> State: Therefore f(x) is differentiable at [x-value] iff [parameters and what they equal for any real number t] for any t ∈ ℝ.

How do you check if f(x) is differentiable at some point a?

> Check continuity (LHL = RHL = f(a)). If not continuous, not differentiable.

> Use the definition of a derivative using x = a to simplify the definition equation.

> Define left-hand function/right-hand function to find a substitute for f(h) and find the LHD/RHD of the simplified definition equation.

> Does LHD = RHD?

• Yes → differentiable.

• No → differentiable.

Why does not continuous → not differentiable?

> Differentiability = smooth slope

> If graph has break/jump/gap, there’s no smooth tangent line to take a derivative from.

How do you find all horizontal tangent lines to the graph of a function?

> Find f’(x).

> Set f’(x) = 0.

> Find roots.

> Use the roots to find corresponding y = f(x) values.

> Write the horizontal tangent lines (the y-values).

How do you find the time that it takes for an object to hit the ground given the initial height, initial velocity, and acceleration?

> Write the position equation: h(t) = h₀ + v₀t + ½ at²

> Plug in the given information.

> Set height equal to the ground level.

> Solve for t.

How do you find the velocity of a particle at a given time point with the given s(t)?

> Differentiate once → velocity.

> Plug in t values,

How do you find the time when the acceleration of a particle is 0 with the given s(t)?

> Differentiate → velocity → acceleration.

> Set acceleration = 0.

> Find t.

How do you find the velocity of a particle with a given s(t) equation and determine if it’s moving left or right?

> Differentiate to get velocity. (velocity is the function here)

> Factor velocity.

> Find roots.

> Make sign chart including intervals and sign of v(t) using factors.

> Interpret, i.e. sign of v(t) = (+) = right, sign of v(t) = (-) = left.

How do you approximate some number x using linearization with f(x) with a certain a?

1. Choose nearby easy value “a.”

2. Find/make “f(a).”

3. Find “f’(a).”

4. Use: “f(x) ≈ f(a) + f’(a)(x - a).”

How do you find the equation of a tangent line to a curve equation at a point (x,y)?

> Implicitly differentiate both sides.

> Evaluate the derivative at the given point to get the slope.

> Use the point-slope form.

How do you find the points on a circle equation such that the tangent line passes through a point?

> Implicitly differentiate the curve → dy/dx = m.

> Write in point-slope form using the point given and m.

> Simplify to obtain an equation relating x and y.

> Use original curve and the equation relating x and y to find x and y.

> State the points of tangency.

How do you find the derivative of the inverse function?

> Find f’(x).

> Find f^-1(x).

> Plug into the formula (f^-1)’(x) = 1/f’[f^-1(x)] by subbing f^-1(x) first and then using f’(x).

How do you determine tangent line from a graph through a given point, dy/dx,and dx/dy?

> Find dy/dx.

> Find dx/dy. (dx/dy = 1/(dy/dx)

> Sub given points x and y into solved equation of dy/dx to get the slope.

> Use the point-slope form to find the tangent line equation.