In calculus, the product rule is a technique employed to determine the derivative of the multiplication of two or more functions. It asserts that the derivative of the product of two functions is the result of the first function and the derivative of the second function added to the second function and the derivative of the first.
| Topic | Problem | Solution |
|---|---|---|
| None | Calculate $\frac{d y}{d x}$. You need not expand … | Given the function \(y=(3x^2+x)(x-x^2)\). |
| None | Differentiate. \[ f(x)=x^{6} \ln 5 x \] | The given function is a product of two functions, \(x^6\) and \(\ln(5x)\). |
| None | Find the derivative of the function $y=(4 x+5)^{4… | Given the function \(y=(4 x+5)^{4}(3 x+1)^{-2}\) |
| None | Find dy/dt. 3) $y=t^{4}\left(t^{5}+3\right)^{4}$ | We are given the function \(y=t^{4}(t^{5}+3)^{4}\) and we are asked to find \(\frac{dy}{dt}\). |
| None | $\begin{array}{l}f(x)=\left(4 x^{2}+6\right)^{6}\… | Given the function \(f(x)=(4x^{2}+6)^{6}(3x^{2}+6)^{11}\), we are asked to find its derivative \(f'… |
| None | Suppose $f(x)=x^{2} p(x)$ for some unknown functi… | Given that $f(x)=x^{2} p(x)$, we can find the derivative of $f(x)$ using the product rule. |
| None | Find $\frac{d y}{d x}$ \[ y=6(\tan x+\sec x)(\tan… | Given the function \(y=6(\tan x+\sec x)(\tan x-\sec x)\) |
| None | a. Apply the Product Rule. Let $u=\left(2 x^{2}+3… | Let \(u=2 x^{2}+3\) and \(v=8 x+5+\frac{1}{x}\) |
| None | Find $y^{\prime}$ by (a) applying the Product Rul… | First, we recognize that the given function can be written in the form \((a+b)(c+d+e)\), where \(a=… |
| None | (b) Let $A(x)=(x-2)^{2}(3 x-4)^{2}$ Find $A^{\pri… | Let \(A(x) = (x-2)^{2}(3x-4)^{2}\) |
| None | 20 Consider $f(x)=\frac{1}{100}(3 x-5)(5 x+21)$ a… | \(f(x) = \frac{1}{100}(3x - 5)(5x + 21)\) |