Find dy/dx y=(4x-7)^5(5x+8)^3

The given problem is asking you to find the derivative with respect to x (dy/dx) of the function y, where y is the product of two functions, namely (4x-7)^5 and (5x+8)^3. Essentially, you are being asked to apply the product rule of differentiation, which is a rule used when taking the derivative of a product of two functions. The problem requires you to use this rule along with the power rule (which helps with taking the derivative of functions raised to a power) to calculate the derivative of the entire expression.

$y = {((4 x - 7))}^{5} {((5 x + 8))}^{3}$

Answer

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Solution:

Step:1

Take the derivative of both sides with respect to $x$ : $\frac{d}{d x} y = \frac{d}{d x} ((4 x - 7)^{5} (5 x + 8)^{3})$

Step:2

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

Step:3

Compute the derivative of the right-hand side.

Step:3.1

Apply the Product Rule: $\frac{d}{d x} [u v] = u \frac{d v}{d x} + v \frac{d u}{d x}$ , where $u = (4 x - 7)^{5}$ and $v = (5 x + 8)^{3}$ .

Step:3.2

Use the Chain Rule which says $\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) g^{'} (x)$ , where $f (x) = x^{3}$ and $g (x) = 5 x + 8$ .

Step:3.2.1

Let $u_{1} = 5 x + 8$ and differentiate $(u_{1})^{3}$ with respect to $u_{1}$ and $5 x + 8$ with respect to $x$ .

Step:3.2.2

Apply the Power Rule: $\frac{d}{d u_{1}} (u_{1}^{n}) = n u_{1}^{n - 1}$ , where $n = 3$ .

Step:3.2.3

Substitute $5 x + 8$ back in place of $u_{1}$ .

Step:3.3

Carry out the differentiation.

Step:3.3.1

Rearrange the terms, placing the constant $3$ before $(4 x - 7)^{5}$ .

Step:3.3.2

Use the Sum Rule: $\frac{d}{d x} (a x + b) = \frac{d}{d x} (a x) + \frac{d}{d x} (b)$ .

Step:3.3.3

Since $5$ is a constant, differentiate $5 x$ with respect to $x$ .

Step:3.3.4

Apply the Power Rule: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ , where $n = 1$ .

Step:3.3.5

Multiply $5$ by $1$ .

Step:3.3.6

The derivative of a constant is zero.

Step:3.3.7

Simplify the expression.

Step:3.3.7.1

Combine $5$ and $0$ .

Step:3.3.7.2

Multiply $5$ by $3$ .

Step:3.3.7.3

Factor out the common term $5 (4 x - 7)^{4} (5 x + 8)^{2}$ .

Step:4

Combine the left and right sides into a single equation.

Step:5

Replace $y$ with $\frac{d y}{d x}$ to complete the differentiation.

Knowledge Notes:

Product Rule: When differentiating a product of two functions, $u (x) v (x)$ , the derivative is $u^{'} (x) v (x) + u (x) v^{'} (x)$ .
Chain Rule: Used to differentiate composite functions. If $y = f (g (x))$ , then $\frac{d y}{d x} = f^{'} (g (x)) g^{'} (x)$ .
Power Rule: For any real number $n$ , the derivative of $x^{n}$ with respect to $x$ is $n x^{n - 1}$ .
Sum Rule: The derivative of a sum of functions is the sum of the derivatives of those functions.
Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
Derivative of a Constant: The derivative of a constant is zero.
Factoring: A common algebraic technique used to simplify expressions and solve equations, which involves finding common factors and grouping terms.