1. **Problem Statement:** Find the values of $a$ and $k$ given that the derivative of $2ax^7 + 3x^k$ is $42x^6 + 15x^{k - 1}$. 2. **Recall the derivative rules:** - The derivative of $x^n$ is $nx^{n-1}$. - The derivative of a sum is the sum of the derivatives. 3. **Differentiate the given function:** $$\frac{d}{dx}(2ax^7 + 3x^k) = 2a \cdot 7x^{6} + 3 \cdot k x^{k-1} = 14ax^6 + 3kx^{k-1}$$ 4. **Set the derivative equal to the given expression:** $$14ax^6 + 3kx^{k-1} = 42x^6 + 15x^{k-1}$$ 5. **Equate coefficients of like terms:** - For $x^6$: $14a = 42$ so $a = \frac{42}{14} = 3$ - For $x^{k-1}$: $3k = 15$ so $k = \frac{15}{3} = 5$ **Final answers:** $$a = 3$$ $$k = 5$$