Find dy/dx y = square root of x(x-2)

Solution:

Step:1

Convert the square root into an exponent using the rule $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$. Thus, $\sqrt{x}$ becomes $x^{\frac{1}{2}}$. The equation now is $y = x^{\frac{1}{2}}(x - 2)$.

Step:2

Take the derivative of both sides with respect to $x$: $\frac{d}{dx}(y) = \frac{d}{dx}\left(x^{\frac{1}{2}}(x - 2)\right)$.

Step:3

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

Step:4

Apply the derivative to the right-hand side of the equation.

Step:4.1

Use the Product Rule: $\frac{d}{dx}[f(x)g(x)] = f(x)\frac{d}{dx}[g(x)] + g(x)\frac{d}{dx}[f(x)]$, where $f(x) = x^{\frac{1}{2}}$ and $g(x) = x - 2$. This gives us $x^{\frac{1}{2}}\frac{d}{dx}[x - 2] + (x - 2)\frac{d}{dx}[x^{\frac{1}{2}}]$.

Step:4.2

Proceed with differentiation.

Step:4.2.1

Apply the Sum Rule: the derivative of $x - 2$ is the sum of the derivatives of its terms. We get $x^{\frac{1}{2}}(1 + 0) + (x - 2)\frac{d}{dx}[x^{\frac{1}{2}}]$.

Step:4.2.2

Use the Power Rule: $\frac{d}{dx}[x^n] = nx^{n-1}$, where $n = 1$. This simplifies to $x^{\frac{1}{2}} + (x - 2)\frac{d}{dx}[x^{\frac{1}{2}}]$.

Step:4.2.3

The derivative of a constant is zero, so the derivative of $-2$ is $0$. We now have $x^{\frac{1}{2}} + (x - 2)\frac{d}{dx}[x^{\frac{1}{2}}]$.

Step:4.2.4

Simplify the expression.

Step:4.2.4.1

Add $1$ and $0$ together to get $x^{\frac{1}{2}} + (x - 2)\frac{d}{dx}[x^{\frac{1}{2}}]$.

Step:4.2.4.2

Multiplying $x^{\frac{1}{2}}$ by $1$ does not change its value.

Step:4.2.5

Apply the Power Rule again with $n = \frac{1}{2}$ to get $x^{\frac{1}{2}} + (x - 2)\left(\frac{1}{2}x^{\frac{1}{2} - 1}\right)$.

Step:4.3

To express $-1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$.

Step:4.4

Combine the terms to get $x^{\frac{1}{2}} + (x - 2)\left(\frac{1}{2}x^{\frac{-1}{2}}\right)$.

Step:4.5

Simplify the expression further.

Step:4.5.1

Multiply $-1$ by $2$.

Step:4.5.2

Subtract $2$ from $1$ to get $x^{\frac{1}{2}} + (x - 2)\left(\frac{1}{2}x^{\frac{-1}{2}}\right)$.

Step:4.6

Move the negative exponent to the denominator using the rule $b^{-n} = \frac{1}{b^n}$.

Step:4.7

Combine the fraction to get $x^{\frac{1}{2}} + (x - 2)\frac{1}{2x^{\frac{1}{2}}}$.

Step:4.8

Simplify by distributing and combining like terms.

Step:4.9

Factor out common terms and cancel where possible.

Step:5

Reformulate the equation by setting the left side equal to the simplified right side: $y = \frac{3x^{\frac{1}{2}}}{2} - \frac{1}{x^{\frac{1}{2}}}$.

Step:6

Replace $y$ with $\frac{dy}{dx}$ to get the final derivative: $\frac{dy}{dx} = \frac{3x^{\frac{1}{2}}}{2} - \frac{1}{x^{\frac{1}{2}}}$.

Knowledge Notes:

• The Product Rule is used when differentiating products of two functions: $\frac{d}{dx}[f(x)g(x)] = f(x)\frac{d}{dx}[g(x)] + g(x)\frac{d}{dx}[f(x)]$.

• The Sum Rule states that the derivative of a sum of functions is the sum of their derivatives.

• The Power Rule for differentiation is given by $\frac{d}{dx}[x^n] = nx^{n-1}$.

• A constant's derivative with respect to any variable is zero.

• Negative exponents indicate reciprocals: $b^{-n} = \frac{1}{b^n}$.

• Simplifying expressions often involves factoring, distributing, and combining like terms.

• The derivative of a function represents the rate at which the function's value changes with respect to changes in the input value.