Solve f^3-5f+25-4f^2-12f+9

$${ f }^{ 3 } -5f+25-4 { f }^{ 2 } -12f+9$$

Evaluate

$$f^{3}-4f^{2}-17f+34$$

Solution Steps

Combine -5f and -12f to get -17f.

$$f^{3}-17f+25-4f^{2}+9$$

Add 25 and 9 to get 34.

$$f^{3}-17f+34-4f^{2}$$

Differentiate w.r.t. f

$$3f^{2}-8f-17$$

Steps Using Derivative Rule for Sum

Combine -5f and -12f to get -17f.

$$\frac{\mathrm{d}}{\mathrm{d}f}(f^{3}-17f+25-4f^{2}+9)$$

Add 25 and 9 to get 34.

$$\frac{\mathrm{d}}{\mathrm{d}f}(f^{3}-17f+34-4f^{2})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$3f^{3-1}-17f^{1-1}+2\left(-4\right)f^{2-1}$$

Subtract 1 from 3.

$$3f^{2}-17f^{1-1}+2\left(-4\right)f^{2-1}$$

Subtract 1 from 1.

$$3f^{2}-17f^{0}+2\left(-4\right)f^{2-1}$$

Multiply 1 times -17.

$$3f^{2}-17f^{0}-8f^{2-1}$$

Subtract 1 from 2.

$$3f^{2}-17f^{0}-8f^{1}$$

For any term t, t^{1}=t.

$$3f^{2}-17f^{0}-8f$$

For any term t except 0, t^{0}=1.

$$3f^{2}-17-8f$$