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Derivative of sin⁡(x3)\sin\left(x^{3}\right)

The derivative of sin⁡(x3)\sin\left(x^{3}\right) is cos⁡(x3)3x2\cos\left(x^{3}\right) 3 x^{2}.

f(x) =
ddx[sin⁡(x3)]=cos⁡(x3)3x2\frac{d}{dx}\left[\sin\left(x^{3}\right)\right] = \cos\left(x^{3}\right) 3 x^{2}

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Graph of f(x) = sin(x^3) and its derivative-3-2-1123-10-551015slope 1.621f(x)f'(x)tangent at x = 1
Graph of f(x) = sin(x^3) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 1.6209.

How to work it out

  1. 1

    We differentiate f(x)=sin⁡(x3)f(x) = \sin\left(x^{3}\right) with respect to x. The rule that applies is the chain rule.

    f(x)=sin⁡(x3)f(x) = \sin\left(x^{3}\right)
  2. 2

    Chain rule: differentiate the outer function, keep the inside, then multiply by the derivative of the inside. The inside is g(x)=x3g(x) = x^{3}.

    g(x)=x3,g′(x)=3x2g(x) = x^{3},\quad g'(x) = 3 x^{2}
  3. 3

    The outer function is sin(u); its derivative evaluated at g(x), times g'(x):

    f′(x)=[cos⁡(u)]u=g(x)⋅g′(x)f'(x) = \left[\cos\left(u\right)\right]_{u=g(x)} \cdot g'(x)
  4. 4

    Simplify:

    f′(x)=cos⁡(x3)3x2f'(x) = \cos\left(x^{3}\right) 3 x^{2}
  5. 5

    Differentiating once more gives the second derivative, which tells you about curvature:

    f′′(x)=−(sin⁡(x3)3x23x2)+cos⁡(x3)6xf''(x) = -\left(\sin\left(x^{3}\right) 3 x^{2} 3 x^{2}\right) + \cos\left(x^{3}\right) 6 x
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\sin\left(x^{3}\right)\right] = \cos\left(x^{3}\right) 3 x^{2} \]
\[ f(x) = \sin\left(x^{3}\right) \]
\[ g(x) = x^{3},\quad g'(x) = 3 x^{2} \]
\[ f'(x) = \left[\cos\left(u\right)\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = \cos\left(x^{3}\right) 3 x^{2} \]
\[ f''(x) = -\left(\sin\left(x^{3}\right) 3 x^{2} 3 x^{2}\right) + \cos\left(x^{3}\right) 6 x \]

Values of f and f' for sin(x3x^{3})

xf(x)f'(x)
-2-0.98935825-1.7460004
-1-0.841470981.6209069
0.50.124674730.74414825
10.841470981.6209069
20.98935825-1.7460004
30.95637593-7.8877478

Questions

What is the derivative of sin(x3x^{3})?

The derivative of sin⁡(x3)\sin\left(x^{3}\right) is cos⁡(x3)3x2\cos\left(x^{3}\right) 3 x^{2}.

What is the second derivative of sin(x3x^{3})?

-(sin(x3x^{3}) * 3 * x2x^{2} * 3 * x2x^{2}) + cos(x3x^{3}) * 6x.

Which rule do you use for sin(x3x^{3})?

The chain rule.

What is the slope of sin(x3x^{3}) at x = 1?

1.6209069.

Related derivatives

    Derivative of sin(x^3): cos(x^3) * 3 * x^2 (with steps) | WordGPT