derivative of f(x)=(2x^2-7)^5-(1+4x^3)^5

Solution

derivative of

f (x) = (2 x^{2} - 7)^{5} - (1 + 4 x^{3})^{5}

- 15360 x^{14} - 15360 x^{11} + 320 x^{9} - 5760 x^{8} - 4480 x^{7} + 22560 x^{5} - 54880 x^{3} - 60 x^{2} + 48020 x

Solution steps

\frac{d}{d x} ((2 x^{2} - 7)^{5} - (1 + 4 x^{3})^{5})

Simplify

(2 x^{2} - 7)^{5} - (1 + 4 x^{3})^{5} : - 1024 x^{15} - 1280 x^{12} + 32 x^{10} - 640 x^{9} - 560 x^{8} + 3760 x^{6} - 13720 x^{4} - 20 x^{3} + 24010 x^{2} - 16808

= \frac{d}{d x} (- 1024 x^{15} - 1280 x^{12} + 32 x^{10} - 640 x^{9} - 560 x^{8} + 3760 x^{6} - 13720 x^{4} - 20 x^{3} + 24010 x^{2} - 16808)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d x} (1024 x^{15}) - \frac{d}{d x} (1280 x^{12}) + \frac{d}{d x} (32 x^{10}) - \frac{d}{d x} (640 x^{9}) - \frac{d}{d x} (560 x^{8}) + \frac{d}{d x} (3760 x^{6}) - \frac{d}{d x} (13720 x^{4}) - \frac{d}{d x} (20 x^{3}) + \frac{d}{d x} (24010 x^{2}) - \frac{d}{d x} (16808)

\frac{d}{d x} (1024 x^{15}) = 15360 x^{14}

\frac{d}{d x} (1280 x^{12}) = 15360 x^{11}

\frac{d}{d x} (32 x^{10}) = 320 x^{9}

\frac{d}{d x} (640 x^{9}) = 5760 x^{8}

\frac{d}{d x} (560 x^{8}) = 4480 x^{7}

\frac{d}{d x} (3760 x^{6}) = 22560 x^{5}

\frac{d}{d x} (13720 x^{4}) = 54880 x^{3}

\frac{d}{d x} (20 x^{3}) = 60 x^{2}

\frac{d}{d x} (24010 x^{2}) = 48020 x

\frac{d}{d x} (16808) = 0

= - 15360 x^{14} - 15360 x^{11} + 320 x^{9} - 5760 x^{8} - 4480 x^{7} + 22560 x^{5} - 54880 x^{3} - 60 x^{2} + 48020 x - 0

Simplify

= - 15360 x^{14} - 15360 x^{11} + 320 x^{9} - 5760 x^{8} - 4480 x^{7} + 22560 x^{5} - 54880 x^{3} - 60 x^{2} + 48020 x