Find dy/dx y = square root of x(x-14)
The given problem is a calculus question asking for the derivative of the function y with respect to x. The function y is defined as the square root of the product x(x-14); mathematically represented as y = √[x(x-14)]. The question requires you to apply differentiation rules to find the derivative dy/dx, which represents the rate at which y changes with respect to changes in x. You would need to use the chain rule and product rule within the differentiation process as the function involves a product inside a square root.
$y = \sqrt{x} \left(\right. x - 14 \left.\right)$
Answer
Solution:
Step 1:
Express the square root as a fractional exponent using the identity $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$.
$y = x^{\frac{1}{2}}(x - 14)$
Step 2:
Take the derivative of both sides with respect to $x$.
$\frac{dy}{dx} = \frac{d}{dx} \left[ x^{\frac{1}{2}}(x - 14) \right]$
Step 3:
The derivative of $y$ with respect to $x$ is denoted as $\frac{dy}{dx}$.
Step 4:
Apply the derivative to the right side of the equation.
Step 4.1:
Utilize the Product Rule: $\frac{d}{dx} [f(x)g(x)] = f(x)\frac{dg(x)}{dx} + g(x)\frac{df(x)}{dx}$, where $f(x) = x^{\frac{1}{2}}$ and $g(x) = x - 14$.
$x^{\frac{1}{2}}\frac{d}{dx}[x - 14] + (x - 14)\frac{d}{dx}[x^{\frac{1}{2}}]$
Step 4.2:
Perform the differentiation.
Step 4.2.1:
Apply the Sum Rule to differentiate $x - 14$.
$x^{\frac{1}{2}}(1 + 0) + (x - 14)\frac{d}{dx}[x^{\frac{1}{2}}]$
Step 4.2.2:
Use the Power Rule, $\frac{d}{dx}[x^{n}] = nx^{n - 1}$, where $n = 1$.
$x^{\frac{1}{2}} + (x - 14)\frac{d}{dx}[x^{\frac{1}{2}}]$
Step 4.2.3:
The derivative of a constant is zero.
$x^{\frac{1}{2}} + (x - 14)\frac{d}{dx}[x^{\frac{1}{2}}]$
Step 4.2.4:
Simplify the expression by adding $1$ and $0$.
$x^{\frac{1}{2}} + (x - 14)\frac{d}{dx}[x^{\frac{1}{2}}]$
Step 4.2.5:
Apply the Power Rule again, with $n = \frac{1}{2}$.
$x^{\frac{1}{2}} + (x - 14)\left(\frac{1}{2}x^{-\frac{1}{2}}\right)$
Step 4.3:
Express $-1$ as a fraction to have a common denominator.
$x^{\frac{1}{2}} + (x - 14)\left(\frac{1}{2}x^{-\frac{1}{2}}\right)$
Step 4.4:
Combine the terms.
$x^{\frac{1}{2}} + (x - 14)\frac{1}{2}x^{-\frac{1}{2}}$
Step 4.5:
Simplify the expression.
$x^{\frac{1}{2}} + \frac{x}{2}x^{-\frac{1}{2}} - \frac{14}{2}x^{-\frac{1}{2}}$
Step 4.6:
Further simplify the expression.
$x^{\frac{1}{2}} + \frac{x^{\frac{1}{2}}}{2} - \frac{7}{x^{\frac{1}{2}}}$
Step 5:
Combine the terms over a common denominator.
$\frac{3x^{\frac{1}{2}}}{2} - \frac{7}{x^{\frac{1}{2}}}$
Step 6:
Set the left side equal to the right side to complete the differentiation.
$\frac{dy}{dx} = \frac{3x^{\frac{1}{2}}}{2} - \frac{7}{x^{\frac{1}{2}}}$
Knowledge Notes:
Square Roots and Exponents: The square root of a number can be expressed as a fractional exponent, where $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$. This is useful for differentiation as it allows us to apply the power rule.
Derivatives: The derivative of a function measures how the function's output changes as its input changes. Notationally, the derivative of $y$ with respect to $x$ is written as $\frac{dy}{dx}$.
Product Rule: When taking the derivative of a product of two functions, the product rule is used: $\frac{d}{dx} [f(x)g(x)] = f(x)\frac{dg(x)}{dx} + g(x)\frac{df(x)}{dx}$.
Sum Rule: The derivative of a sum of functions is the sum of the derivatives: $\frac{d}{dx} [f(x) + g(x)] = \frac{df(x)}{dx} + \frac{dg(x)}{dx}$.
Power Rule: A rule for differentiation that states $\frac{d}{dx}[x^{n}] = nx^{n - 1}$, where $n$ is a real number.
Negative Exponents: A negative exponent indicates that the base is on the opposite side of a fraction. For example, $b^{-n} = \frac{1}{b^{n}}$.
Simplifying Expressions: Combining like terms, factoring, and canceling common factors are all algebraic techniques used to simplify expressions.
