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.
$y = \left(\left(\right. 4 x - 7 \left.\right)\right)^{5} \left(\left(\right. 5 x + 8 \left.\right)\right)^{3}$
Answer
Solution:
Step:1
Take the derivative of both sides with respect to $x$: $\frac{d}{dx} y = \frac{d}{dx} ((4x - 7)^5(5x + 8)^3)$
Step:2
The derivative of $y$ with respect to $x$ is denoted as $\frac{dy}{dx}$.
Step:3
Compute the derivative of the right-hand side.
Step:3.1
Apply the Product Rule: $\frac{d}{dx}[uv] = u\frac{dv}{dx} + v\frac{du}{dx}$, where $u = (4x - 7)^5$ and $v = (5x + 8)^3$.
Step:3.2
Use the Chain Rule which says $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$, where $f(x) = x^3$ and $g(x) = 5x + 8$.
Step:3.2.1
Let $u_1 = 5x + 8$ and differentiate $(u_1)^3$ with respect to $u_1$ and $5x + 8$ with respect to $x$.
Step:3.2.2
Apply the Power Rule: $\frac{d}{du_1}(u_1^n) = nu_1^{n-1}$, where $n = 3$.
Step:3.2.3
Substitute $5x + 8$ 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 $(4x - 7)^5$.
Step:3.3.2
Use the Sum Rule: $\frac{d}{dx}(ax + b) = \frac{d}{dx}(ax) + \frac{d}{dx}(b)$.
Step:3.3.3
Since $5$ is a constant, differentiate $5x$ with respect to $x$.
Step:3.3.4
Apply the Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$, 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 $5(4x - 7)^4(5x + 8)^2$.
Step:4
Combine the left and right sides into a single equation.
Step:5
Replace $y$ with $\frac{dy}{dx}$ 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 $\frac{dy}{dx} = f'(g(x))g'(x)$.
Power Rule: For any real number $n$, the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.
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.
