Power rule for derivatives Calculator

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Example

Created on 2024-06-20Asked by Isabella Lewis (Solvelet student)

Find the derivative of

f(x) = (3x^2 + 2x)^3

with respect to

x

Solution

To find the derivative of

f(x) = (3x^2 + 2x)^3

with respect to

x

: 1. Apply the power rule for derivatives:

\frac{d}{dx}(u^n) = nu^{n-1} \cdot \frac{du}{dx},

where

u = 3x^2 + 2x

and

n = 3

. 2. Compute the derivative:

\begin{aligned} f'(x) &= 3(3x^2 + 2x)^2 \cdot \frac{d}{dx}(3x^2 + 2x) \\ &= 3(3x^2 + 2x)^2 \cdot (6x + 2). \end{aligned}

Therefore, the derivative of

f(x)

with respect to

x

f'(x) = 3(3x^2 + 2x)^2 \cdot (6x + 2)

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Ella Torres on Solvelet

1. Find the derivative of the function

f(x) = 4x^3 - 2x^2 + 5x - 1

.2. Calculate the slope of the tangent line to the curve

y = x^4 - 3x^2 + 2x

at the point

(2, 6)

DefinitionThe power rule for derivatives gives us a fast way to differentiate a function on the form f(x)=xn. For example, it implies that dxd(xn)=nxn−1. For example, if dan_ifn result in f(x)=x3, then f′(x)=3x2.

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