Derivative of y=sin^4x+cos5x+2x^3

1. Differentiate $2 x^{3} + \left(\sin^{4}{\left(x \right)} + \cos{\left(5 x \right)}\right)$ term by term:

   1. Differentiate $\sin^{4}{\left(x \right)} + \cos{\left(5 x \right)}$ term by term:

      1. Let $u = \sin{\left(x \right)}$.

      2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$:

         1. The derivative of sine is cosine:

            $$\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$$

         The result of the chain rule is:

         $$4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}$$

      4. Let $u = 5 x$.

      5. The derivative of cosine is negative sine:

         $$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$$

      6. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$:

         1. The derivative of a constant times a function is the constant times the derivative of the function.

            1. Apply the power rule: $x$ goes to $1$

            So, the result is: $5$

         The result of the chain rule is:

         $$- 5 \sin{\left(5 x \right)}$$

      The result is: $4 \sin^{3}{\left(x \right)} \cos{\left(x \right)} - 5 \sin{\left(5 x \right)}$

   2. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$

      So, the result is: $6 x^{2}$

   The result is: $6 x^{2} + 4 \sin^{3}{\left(x \right)} \cos{\left(x \right)} - 5 \sin{\left(5 x \right)}$

The answer is:

$$6 x^{2} + 4 \sin^{3}{\left(x \right)} \cos{\left(x \right)} - 5 \sin{\left(5 x \right)}$$