Math / Calculus

QuestionDifferentiate the following function. $\begin{array}{c} f(x)=e^{5 x^{2}+9 x} \\ \frac{d}{d x}\left(e^{5 x^{2}+9 x}\right)= \end{array}$ $\square$

Studdy Solution

STEP 1

1. We are given the function $f(x) = e^{5x^2 + 9x}$ .
2. We need to find the derivative of this function with respect to $x$ .

STEP 2

1. Identify the outer and inner functions for differentiation.
2. Apply the chain rule for differentiation.
3. Differentiate the inner function.
4. Combine the results to find the derivative of the original function.

STEP 3

Identify the outer and inner functions:
- Outer function: $e^u$ , where $u = 5x^2 + 9x$ . - Inner function: $u = 5x^2 + 9x$ .

STEP 4

Apply the chain rule for differentiation:
The chain rule states that if $y = g(u)$ and $u = h(x)$ , then:
$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
For our function, this becomes:
$\frac{d}{dx}\left(e^{5x^2 + 9x}\right) = \frac{d}{du}\left(e^u\right) \cdot \frac{du}{dx}$

STEP 5

Differentiate the inner function $u = 5x^2 + 9x$ :
$\frac{du}{dx} = \frac{d}{dx}(5x^2) + \frac{d}{dx}(9x)$
$\frac{du}{dx} = 10x + 9$

STEP 6

Differentiate the outer function $e^u$ with respect to $u$ :
$\frac{d}{du}(e^u) = e^u$
Combine the results using the chain rule:
$\frac{d}{dx}\left(e^{5x^2 + 9x}\right) = e^{5x^2 + 9x} \cdot (10x + 9)$
The derivative of the function is:
$\boxed{e^{5x^2 + 9x} \cdot (10x + 9)}$

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QuestionDifferentiate the following function. $\begin{array}{c} f(x)=e^{5 x^{2}+9 x} \\ \frac{d}{d x}\left(e^{5 x^{2}+9 x}\right)= \end{array}$ $\square$

Studdy Solution

STEP 1

1. We are given the function $f(x) = e^{5x^2 + 9x}$ .
2. We need to find the derivative of this function with respect to $x$ .

STEP 2

1. Identify the outer and inner functions for differentiation.
2. Apply the chain rule for differentiation.
3. Differentiate the inner function.
4. Combine the results to find the derivative of the original function.

STEP 3

Identify the outer and inner functions:
- Outer function: $e^u$ , where $u = 5x^2 + 9x$ . - Inner function: $u = 5x^2 + 9x$ .

STEP 4

Apply the chain rule for differentiation:
The chain rule states that if $y = g(u)$ and $u = h(x)$ , then:
$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
For our function, this becomes:
$\frac{d}{dx}\left(e^{5x^2 + 9x}\right) = \frac{d}{du}\left(e^u\right) \cdot \frac{du}{dx}$

STEP 5

Differentiate the inner function $u = 5x^2 + 9x$ :
$\frac{du}{dx} = \frac{d}{dx}(5x^2) + \frac{d}{dx}(9x)$
$\frac{du}{dx} = 10x + 9$

STEP 6

Differentiate the outer function $e^u$ with respect to $u$ :
$\frac{d}{du}(e^u) = e^u$
Combine the results using the chain rule:
$\frac{d}{dx}\left(e^{5x^2 + 9x}\right) = e^{5x^2 + 9x} \cdot (10x + 9)$
The derivative of the function is:
$\boxed{e^{5x^2 + 9x} \cdot (10x + 9)}$

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QuestionDifferentiate the following function. f(x)=e5x2+9xddx(e5x2+9x)=\begin{array}{c} f(x)=e^{5 x^{2}+9 x} \\ \frac{d}{d x}\left(e^{5 x^{2}+9 x}\right)= \end{array}f(x)=e5x2+9xdxd​(e5x2+9x)=​ □\square□

Studdy Solution

STEP 1

1. We are given the function f(x)=e5x2+9x f(x) = e^{5x^2 + 9x} f(x)=e5x2+9x.2. We need to find the derivative of this function with respect to x x x.

STEP 2

1. Identify the outer and inner functions for differentiation.2. Apply the chain rule for differentiation.3. Differentiate the inner function.4. Combine the results to find the derivative of the original function.

STEP 3

Identify the outer and inner functions:- Outer function: eu e^u eu, where u=5x2+9x u = 5x^2 + 9x u=5x2+9x. - Inner function: u=5x2+9x u = 5x^2 + 9x u=5x2+9x.

STEP 4

STEP 5

Differentiate the inner function u=5x2+9x u = 5x^2 + 9x u=5x2+9x:dudx=ddx(5x2)+ddx(9x) \frac{du}{dx} = \frac{d}{dx}(5x^2) + \frac{d}{dx}(9x) dxdu​=dxd​(5x2)+dxd​(9x)dudx=10x+9 \frac{du}{dx} = 10x + 9 dxdu​=10x+9

STEP 6

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QuestionDifferentiate the following function. $\begin{array}{c} f(x)=e^{5 x^{2}+9 x} \\ \frac{d}{d x}\left(e^{5 x^{2}+9 x}\right)= \end{array}$ $\square$

1. We are given the function $f(x) = e^{5x^2 + 9x}$ .
2. We need to find the derivative of this function with respect to $x$ .

1. Identify the outer and inner functions for differentiation.
2. Apply the chain rule for differentiation.
3. Differentiate the inner function.
4. Combine the results to find the derivative of the original function.

Identify the outer and inner functions:
- Outer function: $e^u$ , where $u = 5x^2 + 9x$ . - Inner function: $u = 5x^2 + 9x$ .

Differentiate the inner function $u = 5x^2 + 9x$ :
$\frac{du}{dx} = \frac{d}{dx}(5x^2) + \frac{d}{dx}(9x)$
$\frac{du}{dx} = 10x + 9$