Conceptual Understanding of Rates of Change A rate is a comparison between two quantities with different units. We often judge performances by using rates. For example: Average speed of a motor vehicle is $90$ km per hour. Average typing rate is $55$ words per minute. Average score of a basketball player is $34$ points per […]

# Differentiation

# When No Need to Apply Quotient Rule for Differentiating a Fraction

The following examples show that you do not need to apply the quotient rule for differentiating when the denominator is a constant. Please see the following cases with the same question. 1. $\textit{Application of the quotient rule}$ \( \begin{align} \displaystyle \dfrac{d}{dx}\dfrac{48x – 4x^3}{3} &= \frac{{\frac{d}{{dx}}\left( {48x – 4{x^3}} \right) \times 3 – \left( {48x – […]

# Logarithmic Differentiation

Share0 Share +10 Tweet0 Our Courses Basic Rule of Logarithmic Differentiation $$ \displaystyle \dfrac{d}{dx}\log_e{x} = \dfrac{1}{x} \\ \dfrac{d}{dx}\log_e{f(x)} = \dfrac{f'(x)}{f(x)} $$ Practice Questions Question 1 Differentiate \( y = \log_{e}(3x) \). \( \begin{aligned} \displaystyle \dfrac{d}{dx}\log_{e}(3x) &= \dfrac{(3x)’}{3x} \\ &= \dfrac{3}{3x} \\ &= \dfrac{1}{x} \end{aligned} \) Question 2 Differentiate \( y = \log_{e}(2x-1) \). \( \begin{aligned} […]

# Implicit Differentiation | Calculus Help

Share0 Share +10 Tweet0 Implicit Differentiation for Calculus Problems This very powerful differentiation process follows from the chain rule. $$u = g(f(x)) \\ \frac{du}{dx} = g'(f(x)) \times f'(x)$$ We’ve done quite a few differentiation and derivatives, but they all have been differentiation of functions of the form \( y = f(x) \). Not all the […]

# 5 Important Patterns of First Principles

Share0 Share +10 Tweet0 First Principles The First Principles defines how basic rule of derivative of a function. $$\displaystyle f'(x) = \lim_{h\to\infty} \frac{f(x+h)-f(x)}{h}$$ Patterns of the First Principles Pattern 1 Differentiate \( f(x) = x^2 \) from the first principles. Pattern 2 Differentiate \( f(x) = x^3 \) from the first principles. Pattern 3 Differentiate […]

# Finding a Function from Differential Equation

Share0 Share +10 Tweet0 The solution of a differential equation is to find an expression without \(\displaystyle \frac{d}{dx} \) notations using given conditions. Note that the proper rules must be in place in order to achieve the valid solution of the differential equations, such as product rule, quotient rule and chain rule particularly. Many students […]