Derivative of $\sin\left(x^{2}\right)$

Q: What is the derivative of sin(x^2)?

The derivative of ≤ft(x^2) is ≤ft(x^2) 2 x.

Q: What is the second derivative of sin(x^2)?

-(sin(x^2) * 2x * 2x) + 2cos(x^2).

The derivative of $\sin\left(x^{2}\right)$ is $\cos\left(x^{2}\right) 2 x$ .

f(x) =

\frac{d}{dx}\left[\sin\left(x^{2}\right)\right] = \cos\left(x^{2}\right) 2 x

Type any function of x; the derivative is computed symbolically in your browser.

\frac{d}{dx}\left[\sin\left(x^{2}\right)\right] = \cos\left(x^{2}\right) 2 x

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Graph of f(x) = sin(x^2) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 1.0806.

How to work it out

1
We differentiate $f(x) = \sin\left(x^{2}\right)$ with respect to x. The rule that applies is the chain rule.
$f(x) = \sin\left(x^{2}\right)$
2
Chain rule: differentiate the outer function, keep the inside, then multiply by the derivative of the inside. The inside is $g(x) = x^{2}$ .
$g(x) = x^{2},\quad g'(x) = 2 x$
3
The outer function is sin(u); its derivative evaluated at g(x), times g'(x):
$f'(x) = \left[\cos\left(u\right)\right]_{u=g(x)} \cdot g'(x)$
4
Simplify:
$f'(x) = \cos\left(x^{2}\right) 2 x$
5
Differentiating once more gives the second derivative, which tells you about curvature:
$f''(x) = -\left(\sin\left(x^{2}\right) 2 x 2 x\right) + 2 \cos\left(x^{2}\right)$

LaTeX source of the formulas on this page

\[ \frac{d}{dx}\left[\sin\left(x^{2}\right)\right] = \cos\left(x^{2}\right) 2 x \]
\[ f(x) = \sin\left(x^{2}\right) \]
\[ g(x) = x^{2},\quad g'(x) = 2 x \]
\[ f'(x) = \left[\cos\left(u\right)\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = \cos\left(x^{2}\right) 2 x \]
\[ f''(x) = -\left(\sin\left(x^{2}\right) 2 x 2 x\right) + 2 \cos\left(x^{2}\right) \]

Values of f and f' for sin( $x^{2}$ )

x	f(x)	f'(x)
-2	-0.7568025	2.6145745
-1	0.84147098	-1.0806046
0.5	0.24740396	0.96891242
1	0.84147098	1.0806046
2	-0.7568025	-2.6145745
3	0.41211849	-5.4667816

Questions

What is the derivative of sin(

x^{2}

The derivative of $\sin\left(x^{2}\right)$ is $\cos\left(x^{2}\right) 2 x$ .

What is the second derivative of sin(

x^{2}

-(sin( $x^{2}$ ) * 2x * 2x) + 2cos( $x^{2}$ ).

Which rule do you use for sin(

x^{2}

The chain rule.

What is the slope of sin(

x^{2}

) at x = 1?

1.0806046.

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How to work it out

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Questions

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Derivative of $\sin\left(x^{2}\right)$

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