Find dy/dx y=( natural log of x)/(x^7)
The problem is asking for the derivative of a given function with respect to x. The function described is y = (ln(x)) / (x^7), where ln(x) represents the natural logarithm of x. To find dy/dx, you need to differentiate the function y with respect to x, which, in this case, will involve applying the quotient rule and the chain rule since the function is a division of two separate functions of x. The quotient rule is a method for differentiating expressions that are fractions of two differentiable functions, while the chain rule is used for differentiating composite functions.
$y = \frac{ln \left(\right. x \left.\right)}{x^{7}}$
Answer
Solution:
Step 1:
Take the derivative of both sides of the equation with respect to $x$: $\frac{d}{dx}(y) = \frac{d}{dx}\left(\frac{\ln(x)}{x^7}\right)$.
Step 2:
The derivative of $y$ with respect to $x$ is denoted as $\frac{dy}{dx}$.
Step 3:
Proceed to differentiate the right-hand side of the equation.
Step 3.1:
Apply the Quotient Rule for differentiation: $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$, where $u = \ln(x)$ and $v = x^7$. This gives us $\frac{x^7\frac{d}{dx}(\ln(x)) - \ln(x)\frac{d}{dx}(x^7)}{(x^7)^2}$.
Step 3.2:
Simplify the denominator by multiplying the exponents in $(x^7)^2$.
Step 3.2.1:
Use the rule for exponents $(a^m)^n = a^{mn}$ to get $\frac{x^7\frac{d}{dx}(\ln(x)) - \ln(x)\frac{d}{dx}(x^7)}{x^{7 \cdot 2}}$.
Step 3.2.2:
Calculate $7 \cdot 2$ to obtain $\frac{x^7\frac{d}{dx}(\ln(x)) - \ln(x)\frac{d}{dx}(x^7)}{x^{14}}$.
Step 3.3:
The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$, leading to $\frac{x^7\cdot\frac{1}{x} - \ln(x)\frac{d}{dx}(x^7)}{x^{14}}$.
Step 3.4:
Differentiate $x^7$ using the Power Rule.
Step 3.4.1:
Combine $x^7$ and $\frac{1}{x}$ to get $\frac{x^6 - \ln(x)\frac{d}{dx}(x^7)}{x^{14}}$.
Step 3.4.2:
Cancel the common factor between $x^7$ and $x$.
Step 3.4.2.1:
Extract $x$ from $x^7$ to get $\frac{x\cdot x^6 - \ln(x)\frac{d}{dx}(x^7)}{x^{14}}$.
Step 3.4.2.2:
Cancel out the common $x$ factors to simplify the expression to $\frac{x^6 - \ln(x)\frac{d}{dx}(x^7)}{x^{14}}$.
Step 3.4.3:
Apply the Power Rule which states that the derivative of $x^n$ is $nx^{n-1}$ where $n = 7$, resulting in $\frac{x^6 - \ln(x)(7x^{6})}{x^{14}}$.
Step 3.4.4:
Factor out the common $x^6$ term.
Step 3.4.4.1:
Multiply $7$ by $-1$ to get $\frac{x^6 - 7\ln(x)x^6}{x^{14}}$.
Step 3.4.4.2:
Factor $x^6$ from the expression to obtain $\frac{x^6(1 - 7\ln(x))}{x^{14}}$.
Step 3.5:
Cancel the $x^6$ term from both the numerator and the denominator.
Step 3.5.1:
Factor $x^6$ from $x^{14}$ to get $\frac{x^6(1 - 7\ln(x))}{x^6x^8}$.
Step 3.5.2:
Cancel the common $x^6$ factor to simplify to $\frac{1 - 7\ln(x)}{x^8}$.
Step 3.6:
Simplify the expression by incorporating the $7$ into the logarithm, resulting in $\frac{1 - \ln(x^7)}{x^8}$.
Step 4:
Express the derivative of $y$ with respect to $x$ as $\frac{dy}{dx} = \frac{1 - \ln(x^7)}{x^8}$.
Knowledge Notes:
Quotient Rule: When differentiating a function that is the quotient of two functions, $f(x) = \frac{u(x)}{v(x)}$, the derivative is given by $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$.
Power Rule: The derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.
Derivative of the Natural Logarithm: The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$.
Simplifying Expressions: When simplifying expressions involving exponents, common factors can be canceled out to simplify the expression further.
Logarithm Properties: The coefficient of a logarithm can be incorporated as the power of the argument, i.e., $a\ln(x) = \ln(x^a)$.
Differentiating Both Sides of an Equation: When differentiating both sides of an equation with respect to $x$, the derivatives must be equal if the original equation is valid.
