Find dy/dx y=8 log base 2 of x

Solution:

Step:1

Apply differentiation to both sides of the given equation: $\frac{d}{dx}(y) = \frac{d}{dx}(8 \log_2(x))$.

Step:2

The derivative of $y$ with respect to $x$ is denoted by $\frac{dy}{dx}$.

Step:3

Proceed to differentiate the right-hand side of the equation.

Step:3.1

Recognize that $8$ is a constant factor and pull it outside the differentiation operator: $8 \cdot \frac{d}{dx}(\log_2(x))$.

Step:3.2

Compute the derivative of $\log_2(x)$ using the change of base formula: $\frac{d}{dx}(\log_2(x)) = \frac{1}{x \ln(2)}$.

Step:3.3

Multiply the constant $8$ by the derivative of the logarithm: $\frac{8}{x \ln(2)}$.

Step:4

Express the derivative of $y$ with respect to $x$ as equal to the result from the previous step: $\frac{dy}{dx} = \frac{8}{x \ln(2)}$.

Step:5

Substitute $\frac{dy}{dx}$ for $y$ in the final expression: $\frac{dy}{dx} = \frac{8}{x \ln(2)}$.

Knowledge Notes:

1. Differentiation: Differentiation is the process of finding the derivative of a function. The derivative represents the rate at which a function is changing at any given point and is a fundamental tool in calculus.

2. Logarithmic Differentiation: When differentiating logarithmic functions, we use the fact that the derivative of $\log_a(x)$ with respect to $x$ is $\frac{1}{x \ln(a)}$, where $a$ is the base of the logarithm and $\ln$ denotes the natural logarithm.

3. Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. Mathematically, if $c$ is a constant and $f(x)$ is a function, then $\frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d}{dx}(f(x))$.

4. Change of Base Formula: The logarithm of a number $x$ to base $a$ can be expressed in terms of logarithms to any other base $b$ using the change of base formula: $\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$. In particular, for natural logarithms, $\log_a(x) = \frac{\ln(x)}{\ln(a)}$.

5. Natural Logarithm: The natural logarithm, denoted as $\ln(x)$, is the logarithm to the base $e$, where $e$ is an irrational and transcendental constant approximately equal to 2.71828. The natural logarithm has properties that make it particularly useful in calculus.

By applying these principles, we can differentiate a variety of functions, including those involving logarithms with different bases.