Find dy/dx y=8^(4x^2+9x)

Solution:

Step:1

Take the derivative of both sides with respect to $x$: $\frac{d}{dx}(y) = \frac{d}{dx}(8^{4x^2+9x})$

Step:2

The derivative of $y$ with respect to $x$ is denoted by $\frac{dy}{dx}$.

Step:3

Compute the derivative of the right-hand side.

Step:3.1

Utilize the chain rule for differentiation: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$, where $f(x) = 8^x$ and $g(x) = 4x^2 + 9x$.

Step:3.1.1

Introduce a substitution $u = 4x^2 + 9x$ and differentiate: $\frac{d}{du}(8^u) \cdot \frac{d}{dx}(4x^2 + 9x)$

Step:3.1.2

Apply the exponential rule: $\frac{d}{du}(a^u) = a^u \ln(a)$, where $a = 8$: $8^u \ln(8) \cdot \frac{d}{dx}(4x^2 + 9x)$

Step:3.1.3

Substitute back $u$ with $4x^2 + 9x$: $8^{4x^2 + 9x} \ln(8) \cdot \frac{d}{dx}(4x^2 + 9x)$

Step:3.2

Proceed with differentiation.

Step:3.2.1

Apply the sum rule: the derivative of $4x^2 + 9x$ is the sum of the derivatives: $8^{4x^2 + 9x} \ln(8) \left(\frac{d}{dx}(4x^2) + \frac{d}{dx}(9x)\right)$

Step:3.2.2

Differentiate $4x^2$ by treating $4$ as a constant: $8^{4x^2 + 9x} \ln(8) \left(4 \frac{d}{dx}(x^2) + \frac{d}{dx}(9x)\right)$

Step:3.2.3

Apply the power rule: $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n = 2$: $8^{4x^2 + 9x} \ln(8) \left(4(2x) + \frac{d}{dx}(9x)\right)$

Step:3.2.4

Combine the constants $2$ and $4$: $8^{4x^2 + 9x} \ln(8) \left(8x + \frac{d}{dx}(9x)\right)$

Step:3.2.5

Differentiate $9x$ by treating $9$ as a constant: $8^{4x^2 + 9x} \ln(8) \left(8x + 9 \frac{d}{dx}(x)\right)$

Step:3.2.6

Apply the power rule for $n = 1$: $8^{4x^2 + 9x} \ln(8) \left(8x + 9 \cdot 1\right)$

Step:3.2.7

Simplify the expression.

Step:3.2.7.1

Multiply $9$ by $1$: $8^{4x^2 + 9x} \ln(8) (8x + 9)$

Step:3.2.7.2

Rearrange the terms: $8^{4x^2 + 9x} (8x + 9) \ln(8)$

Step:4

Combine the left and right sides into the final differentiated form: $\frac{dy}{dx} = 8^{4x^2 + 9x} (8x + 9) \ln(8)$

Step:5

Replace $y$ with $\frac{dy}{dx}$ in the final result: $\frac{dy}{dx} = 8^{4x^2 + 9x} (8x + 9) \ln(8)$

Knowledge Notes:

• Chain Rule: This is a fundamental rule in calculus used to differentiate compositions of functions. If a variable $y$ depends on a variable $u$ which itself depends on $x$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

• Exponential Rule: When differentiating an exponential function of the form $a^u$, where $a$ is a constant and $u$ is a function of $x$, the derivative is $a^u \ln(a) \cdot \frac{du}{dx}$.

• Sum Rule: The derivative of a sum of two functions is the sum of the derivatives of those functions.

• Power Rule: A rule for differentiation that states if $f(x) = x^n$, then $f'(x) = nx^{n-1}$, where $n$ is a real number.

• Constants in Differentiation: When differentiating a term that includes a constant multiplied by a function of $x$, the constant can be factored out and the derivative of the function taken separately.

• Natural Logarithm (ln): The natural logarithm of a number is its logarithm to the base of the mathematical constant $e$, where $e$ is an irrational and transcendental number approximately equal to 2.71828. In this problem, $\ln(8)$ is used in the differentiation of the exponential function.