Find dy/dx y=cos(x)^8
The problem involves finding the first derivative of a function with respect to x. The function in question is y = cos(x)^8, which means that y is equal to the cosine of x, raised to the power of 8. The task is to compute the rate at which y changes as x changes, which is mathematically represented as dy/dx. This involves applying the rules of differentiation to the given trigonometric function.
$y = \left(cos\right)^{8} \left(\right. x \left.\right)$
Answer
Solution:
Step 1:
Take the derivative of both sides of the equation with respect to $x$: $\frac{d}{dx}(y) = \frac{d}{dx}(\cos(x)^8)$.
Step 2:
The derivative of $y$ with respect to $x$ is denoted as $\frac{dy}{dx}$.
Step 3:
Proceed to differentiate the right-hand side.
Step 3.1:
Employ the chain rule for differentiation: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$, where $f(x) = x^8$ and $g(x) = \cos(x)$.
Step 3.1.1:
Introduce a substitution $u = \cos(x)$, then differentiate $u^8$ with respect to $u$ and $\cos(x)$ with respect to $x$: $\frac{d}{du}(u^8) \cdot \frac{d}{dx}(\cos(x))$.
Step 3.1.2:
Apply the power rule for differentiation: $\frac{d}{du}(u^n) = n \cdot u^{n-1}$, where $n = 8$: $8u^7 \cdot \frac{d}{dx}(\cos(x))$.
Step 3.1.3:
Substitute back $\cos(x)$ for $u$: $8(\cos(x))^7 \cdot \frac{d}{dx}(\cos(x))$.
Step 3.2:
The derivative of $\cos(x)$ with respect to $x$ is $-\sin(x)$: $8(\cos(x))^7 \cdot (-\sin(x))$.
Step 3.3:
Combine the constants and simplify: $-8(\cos(x))^7 \cdot \sin(x)$.
Step 4:
Express the derivative of $y$ with respect to $x$ as the result of the differentiation: $\frac{dy}{dx} = -8(\cos(x))^7 \cdot \sin(x)$.
Knowledge Notes:
Derivative: The derivative of a function measures how the function value changes as its input changes. Notationally, the derivative of $y$ with respect to $x$ is written as $\frac{dy}{dx}$.
Chain Rule: A fundamental rule in calculus used to differentiate compositions of functions. It states that if $y = f(g(x))$, then the derivative of $y$ with respect to $x$ is $f'(g(x)) \cdot g'(x)$.
Power Rule: A basic differentiation rule that says if $y = u^n$, where $n$ is a constant, then the derivative of $y$ with respect to $u$ is $\frac{dy}{du} = n \cdot u^{n-1}$.
Trigonometric Functions: Functions like $\sin(x)$ and $\cos(x)$ have well-known derivatives, which are essential in solving many calculus problems. The derivative of $\sin(x)$ is $\cos(x)$, and the derivative of $\cos(x)$ is $-\sin(x)$.
Substitution: In calculus, substitution is often used in conjunction with the chain rule. A new variable, typically $u$, is introduced to simplify the differentiation process.
Simplifying Expressions: After differentiation, it is common to simplify the expression by combining like terms and constants to present the final derivative in its simplest form.
