Find dy/dx y=e^(3x^5)
The given problem is asking to calculate the derivative of the function y with respect to x, where y is defined as the exponential function e raised to the power of a polynomial 3x^5. The notation dy/dx refers to the derivative of y relative to x, which represents the rate at which y changes as x changes. This is a calculus problem involving the application of differentiation rules, specifically the chain rule, to find the slope of the curve of the equation y=e^(3x^5) at any given point x.
$y = e^{3 x^{5}}$
Answer
Solution:
Step:1
Take the derivative of both sides with respect to $x$: $\frac{d}{dx}(y) = \frac{d}{dx}(e^{3x^5})$.
Step:2
The derivative of $y$ with respect to $x$ is denoted as $\frac{dy}{dx}$.
Step:3
Compute the derivative of the exponential function on the right.
Step:3.1
Employ the chain rule for differentiation: $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$, where $f(x) = e^x$ and $g(x) = 3x^5$.
Step:3.1.1
Introduce a substitution $u = 3x^5$ to apply the chain rule: $\frac{d}{du}(e^u) \cdot \frac{d}{dx}(3x^5)$.
Step:3.1.2
Apply the rule for differentiating exponential functions: $\frac{d}{du}(a^u) = a^u \ln(a)$, with $a = e$: $e^u \cdot \frac{d}{dx}(3x^5)$.
Step:3.1.3
Substitute back $u = 3x^5$: $e^{3x^5} \cdot \frac{d}{dx}(3x^5)$.
Step:3.2
Proceed with the differentiation.
Step:3.2.1
Recognize that $3$ is a constant and differentiate $3x^5$: $e^{3x^5} \cdot (3 \cdot \frac{d}{dx}(x^5))$.
Step:3.2.2
Apply the power rule for differentiation: $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n = 5$: $e^{3x^5} \cdot (3 \cdot (5x^4))$.
Step:3.2.3
Combine the constants $5$ and $3$: $e^{3x^5} \cdot (15x^4)$.
Step:3.3
Simplify the expression.
Step:3.3.1
Rearrange the factors: $15e^{3x^5}x^4$.
Step:3.3.2
Finalize the ordering of the terms: $15x^4e^{3x^5}$.
Step:4
Form the equation by equating the left and right sides: $\frac{dy}{dx} = 15x^4e^{3x^5}$.
Step:5
Replace $y$ with $\frac{dy}{dx}$ to complete the differentiation: $\frac{dy}{dx} = 15x^4e^{3x^5}$.
Knowledge Notes:
Chain Rule: A fundamental rule in calculus used for differentiating compositions of functions. It states that if $y = f(g(x))$, then the derivative of $y$ with respect to $x$ is $f'(g(x))g'(x)$.
Exponential Function Differentiation: The derivative of $e^x$ with respect to $x$ is $e^x$. For $a^x$, where $a$ is a constant, the derivative is $a^x \ln(a)$.
Power Rule: A basic differentiation rule that states if $y = x^n$, then the derivative of $y$ with respect to $x$ is $ny^{n-1}$.
Constants in Differentiation: When differentiating an expression, any constant multiplier remains unchanged.
Simplifying Expressions: After applying differentiation rules, it is often necessary to simplify the expression by combining like terms and rearranging factors for clarity.
