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

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Solution:

Step:1

Take the derivative of both sides with respect to $x$ : $\frac{d}{d x} (y) = \frac{d}{d x} (e^{3 x^{5}})$ .

Step:2

The derivative of $y$ with respect to $x$ is denoted as $\frac{d y}{d x}$ .

Step:3

Compute the derivative of the exponential function on the right.

Step:3.1

Employ the chain rule for differentiation: $\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) g^{'} (x)$ , where $f (x) = e^{x}$ and $g (x) = 3 x^{5}$ .

Step:3.1.1

Introduce a substitution $u = 3 x^{5}$ to apply the chain rule: $\frac{d}{d u} (e^{u}) \cdot \frac{d}{d x} (3 x^{5})$ .

Step:3.1.2

Apply the rule for differentiating exponential functions: $\frac{d}{d u} (a^{u}) = a^{u} \ln (a)$ , with $a = e$ : $e^{u} \cdot \frac{d}{d x} (3 x^{5})$ .

Step:3.1.3

Substitute back $u = 3 x^{5}$ : $e^{3 x^{5}} \cdot \frac{d}{d x} (3 x^{5})$ .

Step:3.2

Proceed with the differentiation.

Step:3.2.1

Recognize that $3$ is a constant and differentiate $3 x^{5}$ : $e^{3 x^{5}} \cdot (3 \cdot \frac{d}{d x} (x^{5}))$ .

Step:3.2.2

Apply the power rule for differentiation: $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ , where $n = 5$ : $e^{3 x^{5}} \cdot (3 \cdot (5 x^{4}))$ .

Step:3.2.3

Combine the constants $5$ and $3$ : $e^{3 x^{5}} \cdot (15 x^{4})$ .

Step:3.3

Simplify the expression.

Step:3.3.1

Rearrange the factors: $15 e^{3 x^{5}} x^{4}$ .

Step:3.3.2

Finalize the ordering of the terms: $15 x^{4} e^{3 x^{5}}$ .

Step:4

Form the equation by equating the left and right sides: $\frac{d y}{d x} = 15 x^{4} e^{3 x^{5}}$ .

Step:5

Replace $y$ with $\frac{d y}{d x}$ to complete the differentiation: $\frac{d y}{d x} = 15 x^{4} e^{3 x^{5}}$ .

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 $n y^{n - 1}$ .
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.