Apply the sum rule of differentiation, which states that the derivative of a sum is the sum of the…

Question

Apply the sum rule of differentiation, which states that the derivative of a sum is the sum of the derivatives.

ScanSolve AI Tutor · Accepted Answer

Here are the differentiations for each expression.

1. Differentiate $2x^2 + 3x + 4$ .

Apply the sum rule of differentiation, which states that the derivative of a sum is the sum of the derivatives.
$\frac{d}{dx}(2x^2 + 3x + 4) = \frac{d}{dx}(2x^2) + \frac{d}{dx}(3x) + \frac{d}{dx}(4)$
Apply the power rule $\frac{d}{dx}(ax^n) = anx^{n-1}$ and the constant rule $\frac{d}{dx}(c) = 0$ .
$\frac{d}{dx}(2x^2) = 2 \cdot 2x^{2-1} = 4x$ $\frac{d}{dx}(3x) = 3 \cdot 1x^{1-1} = 3x^0 = 3$ $\frac{d}{dx}(4) = 0$
Combine the results.
$\frac{d}{dx}(2x^2 + 3x + 4) = 4x + 3 + 0 = 4x + 3$
The derivative is $\boxed{4x + 3}$ .

2. Differentiate $\frac{7}{8}x^2 + 4x - 3$ .
Apply the sum/difference rule of differentiation.
$\frac{d}{dx}\left(\frac{7}{8}x^2 + 4x - 3\right) = \frac{d}{dx}\left(\frac{7}{8}x^2\right) + \frac{d}{dx}(4x) - \frac{d}{dx}(3)$
Apply the power rule and the constant rule.
$\frac{d}{dx}\left(\frac{7}{8}x^2\right) = \frac{7}{8} \cdot 2x^{2-1} = \frac{14}{8}x = \frac{7}{4}x$ $\frac{d}{dx}(4x) = 4 \cdot 1x^{1-1} = 4$ $\frac{d}{dx}(3) = 0$
Combine the results.
$\frac{d}{dx}\left(\frac{7}{8}x^2 + 4x - 3\right) = \frac{7}{4}x + 4 - 0 = \frac{7}{4}x + 4$
The derivative is $\boxed{\frac{7}{4}x + 4}$ .

3. Differentiate $\frac{1}{(x^2 - 2)^3}$ .
Rewrite the expression using a negative exponent.
$y = \frac{1}{(x^2 - 2)^3} = (x^2 - 2)^{-3}$
Apply the chain rule, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ .
Let $u = x^2 - 2$ .
Then $y = u^{-3}$ .
Differentiate $y$ with respect to $u$ .
$\frac{dy}{du} = -3u^{-3-1} = -3u^{-4}$
Differentiate $u$ with respect to $x$ .
$\frac{du}{dx} = \frac{d}{dx}(x^2 - 2) = 2x - 0 = 2x$
Multiply the derivatives and substitute $u$ back.
$\frac{dy}{dx} = (-3u^{-4})(2x) = -6x u^{-4}$
Substitute $u = x^2 - 2$ :
$\frac{dy}{dx} = -6x(x^2 - 2)^{-4}$
Rewrite the expression with a positive exponent.
$\frac{dy}{dx} = -\frac{6x}{(x^2 - 2)^4}$
The derivative is $\boxed{-\frac{6x}{(x^2 - 2)^4}}$ .

4. Differentiate $x^2 - 3x + 4$ . The notation $\frac{d}{dx}(x^2 - 3x + 4)$ means to find the derivative of the expression with respect to $x$ .
Apply the sum/difference rule of differentiation.
$\frac{d}{dx}(x^2 - 3x + 4) = \frac{d}{dx}(x^2) - \frac{d}{dx}(3x) + \frac{d}{dx}(4)$
Apply the power rule and the constant rule.
$\frac{d}{dx}(x^2) = 2x^{2-1} = 2x$ $\frac{d}{dx}(3x) = 3 \cdot 1x^{1-1} = 3$ $\frac{d}{dx}(4) = 0$
Combine the results.
$\frac{d}{dx}(x^2 - 3x + 4) = 2x - 3 + 0 = 2x - 3$
The derivative is $\boxed{2x - 3}$ .

Apply the sum rule of differentiation, which states that the derivative of a sum is the sum of the derivatives.

Step-by-step solution

Apply the sum rule of differentiation, which states that the derivative of a sum is the sum of the derivatives.

Apply the power rule $\frac{d}{dx}(ax^n) = anx^{n-1}$ and the constant rule $\frac{d}{dx}(c) = 0$ .

Combine the results.

Apply the sum/difference rule of differentiation.

Apply the power rule and the constant rule.

Combine the results.

Rewrite the expression using a negative exponent.

Apply the chain rule, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ .

Differentiate $y$ with respect to $u$ .

Differentiate $u$ with respect to $x$ .

Multiply the derivatives and substitute $u$ back.

Rewrite the expression with a positive exponent.

Apply the sum/difference rule of differentiation.

Apply the power rule and the constant rule.

Combine the results.

Still stuck on this one?

More Mathematics Questions

Apply the sum rule of differentiation, which states that the derivative of a sum is the sum of the derivatives.

Step-by-step solution

Apply the sum rule of differentiation, which states that the derivative of a sum is the sum of the derivatives.

Apply the power rule ddx(axn)=anxn−1\frac{d}{dx}(ax^n) = anx^{n-1}dxd​(axn)=anxn−1 and the constant rule ddx(c)=0\frac{d}{dx}(c) = 0dxd​(c)=0.

Combine the results.

Apply the sum/difference rule of differentiation.

Apply the power rule and the constant rule.

Combine the results.

Rewrite the expression using a negative exponent.

Apply the chain rule, dydx=dydu⋅dudx\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}dxdy​=dudy​⋅dxdu​.

Differentiate yyy with respect to uuu.

Differentiate uuu with respect to xxx.

Multiply the derivatives and substitute uuu back.

Rewrite the expression with a positive exponent.

Apply the sum/difference rule of differentiation.

Apply the power rule and the constant rule.

Combine the results.

Still stuck on this one?

More Mathematics Questions

Apply the power rule $\frac{d}{dx}(ax^n) = anx^{n-1}$ and the constant rule $\frac{d}{dx}(c) = 0$ .

Apply the chain rule, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ .

Differentiate $y$ with respect to $u$ .

Differentiate $u$ with respect to $x$ .

Multiply the derivatives and substitute $u$ back.