Derivative of function \left( 3x^5 - 5x^3 \right)^{\prime} solved

Let’s derive this function:

\left( 3x^5 - 5x^3 \right)^{\prime} = ?

On the left side, you have the main parts of the solution. On the right side, you have step by step explanation.

\left( 3x^5 - 5x^3 \right)^{\prime} =

Use formula: \left[ f(x) \pm g(x) \right]^{\prime} = f^{\prime}(x) \pm g^{\prime}(x)

= \left( 3x^5 \right)^{\prime} - \left( 5x^3 \right)^{\prime}

= \left( 3x^5 \right)^{\prime} - \left( 5x^3 \right)^{\prime} =

Take the first part: \left( 3 \cdot x^5 \right)^{\prime} =

Use formula: \left[c \cdot f(x) \right]^{\prime} = c \cdot \left[ f(x) \right]^{\prime}

= 3 \cdot \left( x^5 \right)^{\prime}

= 3 \cdot \left( x^5 \right)^{\prime} - \left( 5x^3 \right)^{\prime} =

Take the second part: \left( 5 \cdot x^3 \right)^{\prime} =

Use formula: \left[c \cdot f(x) \right]^{\prime} = c \cdot \left[ f(x) \right]^{\prime}

= 5 \cdot \left( x^3 \right)^{\prime}

= 3 \cdot \left( x^5 \right)^{\prime} - 5 \cdot \left( x^3 \right)^{\prime} =

Let’s derive the first function: \left( x^5 \right)^{\prime} =

Use formula: \left( x^n \right)^{\prime} = n \cdot x^{n-1}

= 5 \cdot x^{5-1} =

= 5 \cdot x^4

= 3 \cdot 5 \cdot x^4 - 5 \cdot \left( x^3 \right)^{\prime} =

Let’s derive the second function: \left( x^3 \right)^{\prime} =

Use formula: \left( x^n \right)^{\prime} = n \cdot x^{n-1}

= 3 \cdot x^{3-1} =

= 3 \cdot x^2

= 3 \cdot 5 \cdot x^4 - 5 \cdot 3 \cdot x^2 =

= 15 \cdot x^4 - 15 \cdot x^2

This problem with the solution is from the book “Marian Olejar: Derivatives with Complete Solutions” from the Solved problems series.