derivative of 10x(x^2+7)^4 — step-by-step solution

1) Let's differentiate:

\frac{d}{d x} (10 x \cdot {(x^{2} + 7)}^{4})

10 \cdot \frac{d}{d x} (x \cdot {(x^{2} + 7)}^{4})

For

U1

x

and

V2

{(x^{2} + 7)}^{4}

, use the product/quotient rule:

\frac{d}{d x} (U1 \cdot V2) = \frac{d}{d x} (U1) \cdot V2 + U1 \cdot \frac{d}{d x} (V2)

2) Let's perform the differentiation

U1

\frac{d}{d x} (x)

1

3) Let's differentiate

V2

\frac{d}{d x} ({(x^{2} + 7)}^{4})

Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{(n - 1)}

with f =

x^{2} + 7

4 \cdot \frac{d}{d x} (x^{2} + 7) \cdot {(x^{2} + 7)}^{3}

4 \cdot (\frac{d}{d x} (x^{2}) + 0) \cdot {(x^{2} + 7)}^{3}

4 \cdot 2 x \cdot {(x^{2} + 7)}^{3}

8 x \cdot {(x^{2} + 7)}^{3}

4) You can evaluate the resulted derivative (

U1 \cdot V2

)':

1 \cdot {(x^{2} + 7)}^{4} + x \cdot 8 x \cdot {(x^{2} + 7)}^{3}

{(x^{2} + 7)}^{4} + x^{2} \cdot 8 \cdot {(x^{2} + 7)}^{3}

5) Then you can resume the derivation:

10 \cdot ({(x^{2} + 7)}^{4} + x^{2} \cdot 8 \cdot {(x^{2} + 7)}^{3})

10 \cdot {(x^{2} + 7)}^{4} + 80 x^{2} \cdot {(x^{2} + 7)}^{3}

▶Derivative:

80 x^{2} \cdot {(x^{2} + 7)}^{3} + 10 \cdot {(x^{2} + 7)}^{4}