Find dy/dx y=(3x+5)^10

Solution:

Step 1:

Take the derivative of both sides of the given equation with respect to $x$: $\frac{d}{dx}(y) = \frac{d}{dx}((3x+5)^{10})$.

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:

Utilize the chain rule for differentiation, which is expressed as $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$, where $f(x) = x^{10}$ and $g(x) = 3x + 5$.

Step 3.1.1:

Introduce $u = 3x + 5$ to simplify the differentiation process: $\frac{d}{du}(u^{10}) \cdot \frac{d}{dx}(3x + 5)$.

Step 3.1.2:

Apply the power rule of differentiation, which states that $\frac{d}{du}(u^n) = nu^{n-1}$ for $n = 10$: $10u^{9} \cdot \frac{d}{dx}(3x + 5)$.

Step 3.1.3:

Substitute $u$ back with $3x + 5$: $10(3x + 5)^{9} \cdot \frac{d}{dx}(3x + 5)$.

Step 3.2:

Differentiate the expression.

Step 3.2.1:

Employ the sum rule for differentiation, where the derivative of a sum is the sum of the derivatives: $10(3x + 5)^{9}(\frac{d}{dx}(3x) + \frac{d}{dx}(5))$.

Step 3.2.2:

Recognize that $3$ is a constant multiplier and differentiate $3x$ accordingly: $10(3x + 5)^{9}(3 \cdot \frac{d}{dx}(x) + \frac{d}{dx}(5))$.

Step 3.2.3:

Using the power rule where $\frac{d}{dx}(x^n) = nx^{n-1}$ for $n = 1$: $10(3x + 5)^{9}(3 \cdot 1 + \frac{d}{dx}(5))$.

Step 3.2.4:

Multiply $3$ by $1$: $10(3x + 5)^{9}(3 + \frac{d}{dx}(5))$.

Step 3.2.5:

Since $5$ is a constant, its derivative is zero: $10(3x + 5)^{9}(3 + 0)$.

Step 3.2.6:

Simplify the derivative expression.

Step 3.2.6.1:

Combine $3$ and $0$: $10(3x + 5)^{9} \cdot 3$.

Step 3.2.6.2:

Multiply $3$ by $10$ to obtain the final derivative: $30(3x + 5)^{9}$.

Step 4:

Express the derivative of $y$ with respect to $x$ by equating the left side to the simplified right side: $\frac{dy}{dx} = 30(3x + 5)^{9}$.

Knowledge Notes:

The problem involves finding the derivative of a function of the form $y = (3x+5)^{10}$ with respect to $x$. The solution employs several fundamental rules of differentiation:

1. Chain Rule: This rule is used when differentiating composite functions. If $y = f(g(x))$, then the derivative of $y$ with respect to $x$ is $f'(g(x)) \cdot g'(x)$.

2. Power Rule: This rule states that if $y = u^n$, where $n$ is a constant, then the derivative of $y$ with respect to $u$ is $ny^{n-1}$.

3. Sum Rule: This rule is applied when differentiating a sum of functions. The derivative of a sum is equal to the sum of the derivatives.

4. Constant Multiple Rule: When a function is multiplied by a constant, the derivative of the function is also multiplied by the same constant.

5. Derivative of a Constant: The derivative of a constant is zero.

In the given solution, the chain rule is applied first to differentiate the composite function $(3x+5)^{10}$. The inner function $g(x) = 3x + 5$ is differentiated, and the outer function $f(u) = u^{10}$ is differentiated with respect to $u$. The power rule is then used to differentiate $u^{10}$, and the constant multiple rule is applied to differentiate $3x$. The sum rule is used to separate the derivative of $3x$ and the constant $5$. Finally, the derivative of the constant $5$ is taken, which is zero, and the expression is simplified to obtain the final result.