If \( f(x)=\left(x^{2}+5 x+3\right)^{4} \), then \( f^{\prime}(x)=\square \) \( f^{\prime}(5)=\square \)

Find the first order derivative with respect to \( x \) for \( (x^2+5x+3)^4 \). Evaluate the derivative by following steps: - step0: Evaluate the derivative: \(\frac{d}{dx}\left(\left(x^{2}+5x+3\right)^{4}\right)\) - step1: Use differentiation rules: \(\frac{d}{dg}\left(g^{4}\right)\times \frac{d}{dx}\left(x^{2}+5x+3\right)\) - step2: Find the derivative: \(4g^{3}\times \frac{d}{dx}\left(x^{2}+5x+3\right)\) - step3: Calculate: \(4g^{3}\left(2x+5\right)\) - step4: Substitute back: \(4\left(x^{2}+5x+3\right)^{3}\left(2x+5\right)\) - step5: Evaluate the power: \(4\left(x^{6}+215x^{3}+27+15x^{5}+84x^{4}+252x^{2}+135x\right)\left(2x+5\right)\) - step6: Multiply the terms: \(\left(4x^{6}+860x^{3}+108+60x^{5}+336x^{4}+1008x^{2}+540x\right)\left(2x+5\right)\) - step7: Apply the distributive property: \(4x^{6}\times 2x+4x^{6}\times 5+860x^{3}\times 2x+860x^{3}\times 5+108\times 2x+108\times 5+60x^{5}\times 2x+60x^{5}\times 5+336x^{4}\times 2x+336x^{4}\times 5+1008x^{2}\times 2x+1008x^{2}\times 5+540x\times 2x+540x\times 5\) - step8: Multiply the terms: \(8x^{7}+20x^{6}+1720x^{4}+4300x^{3}+216x+540+120x^{6}+300x^{5}+672x^{5}+1680x^{4}+2016x^{3}+5040x^{2}+1080x^{2}+2700x\) - step9: Add the terms: \(8x^{7}+140x^{6}+3400x^{4}+6316x^{3}+2916x+540+972x^{5}+6120x^{2}\) Substitute \( x=5 \) into the expression \( 8x^{7}+140x^{6}+3400x^{4}+6316x^{3}+2916x+540+972x^{5}+6120x^{2} \). Evaluate the expression by following steps: - step0: Evaluate: \(8x^{7}+140x^{6}+3400x^{4}+6316x^{3}+2916x+540+972x^{5}+6120x^{2}\) - step1: Substitute: \(8\times 5^{7}+140\times 5^{6}+3400\times 5^{4}+6316\times 5^{3}+2916\times 5+540+972\times 5^{5}+6120\times 5^{2}\) - step2: Multiply the terms: \(8\times 5^{7}+2187500+3400\times 5^{4}+6316\times 5^{3}+2916\times 5+540+972\times 5^{5}+6120\times 5^{2}\) - step3: Multiply the terms: \(8\times 5^{7}+2187500+2125000+6316\times 5^{3}+2916\times 5+540+972\times 5^{5}+6120\times 5^{2}\) - step4: Multiply the terms: \(8\times 5^{7}+2187500+2125000+789500+2916\times 5+540+972\times 5^{5}+6120\times 5^{2}\) - step5: Multiply the numbers: \(8\times 5^{7}+2187500+2125000+789500+14580+540+972\times 5^{5}+6120\times 5^{2}\) - step6: Multiply the terms: \(8\times 5^{7}+2187500+2125000+789500+14580+540+3037500+6120\times 5^{2}\) - step7: Multiply the terms: \(8\times 5^{7}+2187500+2125000+789500+14580+540+3037500+153000\) - step8: Add the numbers: \(36\times 5^{7}+2125000+789500+14580+540+3037500+153000\) - step9: Add the numbers: \(4937500+789500+14580+540+3037500+153000\) - step10: Add the numbers: \(8932620\) The derivative of \( f(x) = (x^{2}+5x+3)^{4} \) is \( f'(x) = 8x^{7} + 140x^{6} + 3400x^{4} + 6316x^{3} + 2916x + 540 + 972x^{5} + 6120x^{2} \). Substitute \( x = 5 \) into the derivative expression, we get \( f'(5) = 8932620 \).