Find dy/dx y=5+2x^3-7^x

$y = 5 + 2 x^{3} - 7^{x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (5 + 2 x^{3} - 7^{x})$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

Differentiate.

By the Sum Rule, the derivative of $5 + 2 x^{3} - 7^{x}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 7^{x}]$ .

$\frac{d}{d x} [5] + \frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 7^{x}]$

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 7^{x}]$

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

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

$0 + 2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 7^{x}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$0 + 2 (3 x^{2}) + \frac{d}{d x} [- 7^{x}]$

Multiply $3$ by $2$ .

$0 + 6 x^{2} + \frac{d}{d x} [- 7^{x}]$

Evaluate $\frac{d}{d x} [- 7^{x}]$ .

Since $- 1$ is constant with respect to $x$ , the derivative of $- 7^{x}$ with respect to $x$ is $- \frac{d}{d x} [7^{x}]$ .

$0 + 6 x^{2} - \frac{d}{d x} [7^{x}]$

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $7$ .

$0 + 6 x^{2} - 7^{x} \ln (7)$

Add $0$ and $6 x^{2}$ .

$6 x^{2} - 7^{x} \ln (7)$

Step 4

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

$y' = 6 x^{2} - 7^{x} \ln (7)$

Step 5

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

$\frac{d y}{d x} = 6 x^{2} - 7^{x} \ln (7)$

Find dy/dx y=5+2x^3-7^x

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