Solve d^3/d^3x(xsin5x)

$$\frac { d ^ { 3 } } { d ^ { 3 } x } ( x \sin 5 x )$$

Differentiate w.r.t. x

$$5\cos(5x)$$

Steps Using Chain Rule

Cancel out d^{3} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x}x\sin(5x))$$

Cancel out x and x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\sin(5x))$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$\cos(5x^{1})\frac{\mathrm{d}}{\mathrm{d}x}(5x^{1})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$\cos(5x^{1})\times 5x^{1-1}$$

Simplify.

$$5\cos(5x^{1})$$

For any term t, t^{1}=t.

$$5\cos(5x)$$

Evaluate

$$\sin(5x)$$