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} \displaystyle \dfrac{d}{dx}\log_{e}(2x-1) &= \dfrac{(2x-1)’}{2x-1} […]

# Differentiation

# Implicit Differentiation | Calculus Help

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 functions will fall into […]

# 5 Important Patterns of First Principles

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 \( f(x) = \sqrt{x} […]

# Finding a Function from Differential Equation

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 missed to apply the […]