Řešení - Derivative

Applying the product rule of derivatives.

$$-\sin \left(15x^5\right)\times \frac{d}{dx}\left[15x^5\right]=-\sin \left(15x^5\right)\left(\frac{d}{dx}\left[15\right]\times x^5+15\times \frac{d}{dx}\left[x^5\right]\right)$$

The derivative of a constant value is always zero.

$$-\sin \left(15x^5\right)\left(\frac{d}{dx}\left[15\right]\times x^5+15\times \frac{d}{dx}\left[x^5\right]\right)=-\sin \left(15x^5\right)\left(0x^5+15\times \frac{d}{dx}\left[x^5\right]\right)$$

Multiplying a number by zero always results in zero.

$$-\sin \left(15x^5\right)\left(0x^5+15\times \frac{d}{dx}\left[x^5\right]\right)=-\sin \left(15x^5\right)\left(0+15\times \frac{d}{dx}\left[x^5\right]\right)$$

Adding zero to a number, which does not change its value.

$$-\sin \left(15x^5\right)\left(0+15\times \frac{d}{dx}\left[x^5\right]\right)=-\sin \left(15x^5\right)\times \left(15\times \frac{d}{dx}\left[x^5\right]\right)$$

Computing the derivative of x raised to the power of n.

$$-\sin \left(15x^5\right)\times \left(15\times \frac{d}{dx}\left[x^5\right]\right)=-\sin \left(15x^5\right)\times \left(15\times \left(5x^{5-1}\right)\right)$$

Subtracting one from a number.

$$-\sin \left(15x^5\right)\times \left(15\times \left(5x^{5-1}\right)\right)=-\sin \left(15x^5\right)\times \left(15\times \left(5x^4\right)\right)$$

Multiplication can be grouped differently, but the result remains the same.

$$-\sin \left(15x^5\right)\times \left(15\times \left(5x^4\right)\right)=-\sin \left(15x^5\right)\times \left(\left(15\times 5\right)\times x^4\right)$$

Multiplying two integers together.

$$-\sin \left(15x^5\right)\times \left(\left(15\times 5\right)\times x^4\right)=-\sin \left(15x^5\right)\times \left(75x^4\right)$$

Simplifying the arithmetic expressions.

$$-\sin \left(15x^5\right)\times \left(75x^4\right)=-75x^4\sin \left(15x^5\right)$$