Step-by-step Solution

Find the derivative using the quotient rule $\frac{d}{dx}\left(\frac{\left(x^5+3x\right)^4}{\cos\left(x\right)}\right)$

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Step-by-step solution

Problem to solve:

$\frac{d}{dx}\left(\frac{\left(x^5+3x\right)^4}{cos\:x}\right)$

Solving method

Learn how to solve quotient rule of differentiation problems step by step online.

$\frac{\frac{d}{dx}\left(\left(x^5+3x\right)^4\right)\cos\left(x\right)-\left(x^5+3x\right)^4\frac{d}{dx}\left(\cos\left(x\right)\right)}{\cos\left(x\right)^2}$

Unlock this full step-by-step solution!

Learn how to solve quotient rule of differentiation problems step by step online. Find the derivative using the quotient rule (d/dx)(((x^5+3x)^4)/(cos(x)). Apply the quotient rule for differentiation, which states that if f(x) and g(x) are functions and h(x) is the function defined by {\displaystyle h(x) = \frac{f(x)}{g(x)}}, where {g(x) \neq 0}, then {\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}. The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if f(x) = \cos(x), then f'(x) = -\sin(x)\cdot D_x(x). The derivative of a sum of two functions is the sum of the derivatives of each function.

Final Answer

$\frac{4\left(x^5+3x\right)^{3}\left(5x^{4}+3\right)\cos\left(x\right)+\left(x^5+3x\right)^4\sin\left(x\right)}{\cos\left(x\right)^2}$