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

The given problem is a calculus question asking for the derivative of a function with respect to x. The function provided is an algebraic expression raised to the 10th power. To find dy/dx of y = (3x+5)^10, one would need to apply the rules of differentiation, specifically the chain rule, which is used for finding the derivative of composite functions.

$y = {((3 x + 5))}^{10}$

Answer

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Solution:

Step 1:

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

Step 2:

The derivative of $y$ with respect to $x$ is denoted as $\frac{d y}{d x}$ .

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 $\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)$ , where $f (x) = x^{10}$ and $g (x) = 3 x + 5$ .

Step 3.1.1:

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

Step 3.1.2:

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

Step 3.1.3:

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

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

Step 3.2.2:

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

Step 3.2.3:

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

Step 3.2.4:

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

Step 3.2.5:

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

Step 3.2.6:

Simplify the derivative expression.

Step 3.2.6.1:

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

Step 3.2.6.2:

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

Step 4:

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

Knowledge Notes:

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

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 $f^{'} (g (x)) \cdot g^{'} (x)$ .
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 $n y^{n - 1}$ .
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.
Constant Multiple Rule: When a function is multiplied by a constant, the derivative of the function is also multiplied by the same constant.
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 $(3 x + 5)^{10}$ . The inner function $g (x) = 3 x + 5$ 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 $3 x$ . The sum rule is used to separate the derivative of $3 x$ 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.