Find dy/dx y=-3e^(4x^2)+7

$y = - 3 e^{4 x^{2}} + 7$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (- 3 e^{4 x^{2}} + 7)$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

By the Sum Rule, the derivative of $- 3 e^{4 x^{2}} + 7$ with respect to $x$ is $\frac{d}{d x} [- 3 e^{4 x^{2}}] + \frac{d}{d x} [7]$ .

$\frac{d}{d x} [- 3 e^{4 x^{2}}] + \frac{d}{d x} [7]$

Evaluate $\frac{d}{d x} [- 3 e^{4 x^{2}}]$ .

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 e^{4 x^{2}}$ with respect to $x$ is $- 3 \frac{d}{d x} [e^{4 x^{2}}]$ .

$- 3 \frac{d}{d x} [e^{4 x^{2}}] + \frac{d}{d x} [7]$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 4 x^{2}$ .

To apply the Chain Rule, set $u$ as $4 x^{2}$ .

$- 3 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [4 x^{2}]) + \frac{d}{d x} [7]$

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$- 3 (e^{u} \frac{d}{d x} [4 x^{2}]) + \frac{d}{d x} [7]$

Replace all occurrences of $u$ with $4 x^{2}$ .

$- 3 (e^{4 x^{2}} \frac{d}{d x} [4 x^{2}]) + \frac{d}{d x} [7]$

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$- 3 (e^{4 x^{2}} (4 \frac{d}{d x} [x^{2}])) + \frac{d}{d x} [7]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- 3 (e^{4 x^{2}} (4 (2 x))) + \frac{d}{d x} [7]$

Multiply $2$ by $4$ .

$- 3 (e^{4 x^{2}} (8 x)) + \frac{d}{d x} [7]$

Multiply $8$ by $- 3$ .

$- 24 e^{4 x^{2}} x + \frac{d}{d x} [7]$

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$- 24 e^{4 x^{2}} x + 0$

Simplify.

Add $- 24 e^{4 x^{2}} x$ and $0$ .

$- 24 e^{4 x^{2}} x$

Reorder factors in $- 24 e^{4 x^{2}} x$ .

$- 24 x e^{4 x^{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - 24 x e^{4 x^{2}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - 24 x e^{4 x^{2}}$

Find dy/dx y=-3e^(4x^2)+7

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